{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2026-04-06T14:01:22.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2026-04-06T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":59721,"title":"finding vector pair with min angle between them","description":"given a matrix with more than one row , compare row vectors of the given matrix and find the pair with the minumum angle between them , \"without using the dot fucntion\" \r\nyou can find the angle from the following formula\r\n θ = cos-1 [ (a. b) / (|a| |b|) ]\r\nthe product between the two vectors is the dot product \r\na⋅b=∑(ai​)*(bi)   from i=1 to n​\r\nthe length of a vector is the square root of the sum of the squares of the components\r\n","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 222px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 111px; transform-origin: 407px 111px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003egiven a matrix with more than one row , compare row vectors of the given matrix and find the pair with the minumum angle between them , \"without using the dot fucntion\" \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eyou can find the angle from the following formula\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e θ = cos-1 [ (a. b) / (|a| |b|) ]\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ethe product between the two vectors is the dot product \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003ea\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e⋅\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eb\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e∑(\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ea\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ei\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e​)*(\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eb\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ei)   from i=1 to n\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ethe length of a vector is the square root of the sum of the squares of the components\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [v1 v2] = minAngle(mat)\r\n   \r\nend","test_suite":"%%\r\nmat = [ 155    12   344   356\r\n   305   234   135    40\r\n   227    13   316   310\r\n   266    72    51   178\r\n     8   195   166   289];\r\nres1=[155    12   344   356];\r\nres2=[227    13   316   310];\r\n[v1 v2]=minAngle(mat);\r\nassert(isequal([v1 v2],[res1 res2]))\r\n\r\n%%\r\nmat=[  2    50    47\r\n    24    99    24\r\n    89    23    42\r\n    13    37     9\r\n    66     5    83\r\n     2     5    95\r\n    60    28     3\r\n    52    84    13\r\n    87    35    14\r\n    37    93    17\r\n    80    43   100\r\n    20    16    98\r\n    27    41     3\r\n    59    86    76\r\n    22    56    26\r\n    99    68    42\r\n    13    94    40];\r\n\r\n\r\n \r\n\r\n\r\n\r\n\r\n res1=[   13    37     9];\r\n  res2=[  37    93    17];\r\n [v1 v2]=minAngle(mat);\r\nassert(isequal([v1 v2],[res1 res2]))\r\n\r\n%%\r\nmat=[ 79     4    66    79    23\r\n    22    42    94     3    25\r\n    78    36    58    16    71\r\n    55    52    23    23     8\r\n    67     3     4    80    82\r\n    96    74    85    74    98];\r\n\r\n   res1=[ 78    36    58    16    71];\r\n\r\n\r\n\r\n   res2=[ 96    74    85    74    98];\r\n    [v1 v2]=minAngle(mat);\r\nassert(isequal([v1 v2],[res1 res2]))\r\n\r\n %%\r\n  mat=[ 8    47    50    69\r\n    78    49    11    83\r\n    68    53    63    98\r\n    97    74   100    57\r\n     4    48    24    91\r\n    96    46    12    92\r\n    64    61     4     1\r\n    73    61    72    71\r\n    78     6    47    63\r\n    52    34    18    41\r\n    46     3    95    38\r\n   100    10    52     2\r\n    88    71    59    25\r\n    63    33    99    43\r\n    95    56    48    44\r\n    73     6    13    14\r\n    56    60     8    84\r\n    96    60     7    85\r\n     8    90    54    76\r\n     1     6    39    70\r\n    83    12    18    44\r\n    44     9    85    11\r\n    65    43    17    90\r\n    44    68    36    34\r\n    92    89    46    89\r\n    36    37    65    79\r\n    11    69    76    41\r\n    41    17    23    56\r\n    94    10    16     7\r\n    51    98    17    70\r\n    94    18    53    11\r\n    45     9    38    42\r\n    59    99    32    31\r\n    81    85    12    82\r\n    46    48    87   100\r\n    52    17    78    17\r\n    35    55    61    57\r\n    46    55    22    93\r\n    52    51    31    10\r\n    84    56    31    34\r\n    77    51    62    66\r\n    41    88    17    40\r\n    90    62    86    20\r\n    86    50    60    48\r\n    11    50     3    54\r\n    31    52     5    71\r\n     1     3    98    47\r\n    28    86    93    55\r\n    71    55    16    93\r\n    84     1    68     7\r\n    17    64    98    33\r\n    76    19     2    72\r\n    72     1    56    73\r\n    13    92    67    14\r\n    49     9    92    37\r\n    44    28    83    76\r\n    56    78    58    20\r\n     6     3    75    90\r\n    96    21    63    10\r\n    59     2    60    72\r\n    17    65    28    98\r\n    34    46     5    41\r\n    48    56    54    47\r\n    76    95    90     8\r\n    57    35     5    54\r\n    26    30    13    73\r\n     6    49    17    46\r\n    23    59    82    56\r\n    55    46    46    33\r\n    80    15    42     6\r\n    78    57    59    22\r\n    96    11    69    23\r\n    72    58   100    42\r\n    88    48    23    23\r\n    16    35    48    32\r\n    11    45    79    61\r\n    95    82    31    63\r\n    19    49    73    19\r\n    65    14    81    41\r\n    12    81     4    30\r\n    70    24    33    82\r\n     7     5    66    14\r\n    98    79     3    37\r\n     9    81    91    72\r\n    50    62    99    11\r\n     9    66    18    58\r\n    35    38    29    61\r\n    36     9    44     5\r\n    98    25    92    84\r\n    53     9    33    18\r\n    85    80    93     6\r\n    60    56    25    10\r\n     9    29    90    32\r\n    36    98    38    84\r\n    26    64    21    21\r\n    52    36    58    48\r\n    92    15    35    18\r\n    62    37    98    67\r\n    79    53    24     9\r\n    12    62   100    65\r\n    30    26    24    61\r\n    84    12    29    17];\r\n     res1= [36    37    65    79];\r\n\r\n\r\n\r\n\r\n  res2=[ 46    48    87   100];\r\n  [v1 v2]=minAngle(mat);\r\nassert(isequal([v1 v2],[res1 res2]))\r\n%%\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":5,"created_by":3838707,"edited_by":3838707,"edited_at":"2024-03-28T08:31:50.000Z","deleted_by":null,"deleted_at":null,"solvers_count":8,"test_suite_updated_at":"2024-03-28T08:31:50.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2024-03-27T20:22:32.000Z","updated_at":"2026-01-06T16:39:24.000Z","published_at":"2024-03-27T20:23:40.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003egiven a matrix with more than one row , compare row vectors of the given matrix and find the pair with the minumum angle between them , \\\"without using the dot fucntion\\\" \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eyou can find the angle from the following formula\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e θ = cos-1 [ (a. b) / (|a| |b|) ]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ethe product between the two vectors is the dot product \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e⋅\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eb\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e=\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e∑(\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ei\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e​)*(\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eb\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ei)   from i=1 to n\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e​\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ethe length of a vector is the square root of the sum of the squares of the components\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":44372,"title":"Polarisation","description":"You have n polarising filters stacked one on top of another, and you know each axis angle. How much light gets passed through the filter bank? For more information, see \u003chttps://en.wikipedia.org/wiki/Polarizer Polarizer (Wikipedia)\u003e.\r\n\r\n    \u003e\u003e n = [0, 90];\r\n    \u003e\u003e polarised([0, 90])\r\n\r\n    ans = 0","description_html":"\u003cp\u003eYou have n polarising filters stacked one on top of another, and you know each axis angle. How much light gets passed through the filter bank? For more information, see \u003ca href = \"https://en.wikipedia.org/wiki/Polarizer\"\u003ePolarizer (Wikipedia)\u003c/a\u003e.\u003c/p\u003e\u003cpre\u003e    \u0026gt;\u0026gt; n = [0, 90];\r\n    \u0026gt;\u0026gt; polarised([0, 90])\u003c/pre\u003e\u003cpre\u003e    ans = 0\u003c/pre\u003e","function_template":"function y = polarised(x)\r\n  y = max(x);\r\nend","test_suite":"%%\r\nx = 5;\r\ny_correct = 0.5;\r\nassert(abs(polarised(x)-y_correct) \u003c 1e-10)\r\n%%\r\nx = 180;\r\ny_correct = 0.5;\r\nassert(abs(polarised(x)-y_correct) \u003c 1e-10)\r\n%%\r\nx = 365;\r\ny_correct = 0.5;\r\nassert(abs(polarised(x)-y_correct) \u003c 1e-10)\r\n%%\r\nx = [91, 1];\r\ny_correct = 0;\r\nassert(abs(polarised(x)-y_correct) \u003c 1e-10)\r\n%%\r\na = randi([-360, 360]);\r\nb = 90*(1+2*randi([-3, 3]));\r\nx = [a, a+b];\r\ny_correct = 0;\r\nassert(abs(polarised(x)-y_correct) \u003c 1e-10)\r\n%%\r\na = randi([-360, 360]);\r\nb = 90*(1+2*randi([-3, 3]));\r\nx = [a, a+b];\r\ny_correct = 0;\r\nassert(abs(polarised(x)-y_correct) \u003c 1e-10)\r\n%%\r\nx = [0, 22.5];\r\ny_correct = 0.85355339059/2;\r\nassert(abs(polarised(x)-y_correct) \u003c 1e-10)\r\n%%\r\nx = [0, -45];\r\ny_correct = 0.25;\r\nassert(abs(polarised(x)-y_correct) \u003c 1e-10)\r\n%%\r\nx = [5, 140];\r\ny_correct = 0.25;\r\nassert(abs(polarised(x)-y_correct) \u003c 1e-10)\r\n%%\r\nx = 5 + (1:5)*22.5;\r\ny_correct = 0.53079004294/2;\r\nassert(abs(polarised(x)-y_correct) \u003c 1e-10)\r\n\r\n\r\n\r\n","published":true,"deleted":false,"likes_count":4,"comments_count":10,"created_by":14358,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":269,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":34,"created_at":"2017-10-10T21:58:52.000Z","updated_at":"2026-03-26T15:36:11.000Z","published_at":"2017-10-16T01:45:10.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYou have n polarising filters stacked one on top of another, and you know each axis angle. How much light gets passed through the filter bank? For more information, see\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Polarizer\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePolarizer (Wikipedia)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[    \u003e\u003e n = [0, 90];\\n    \u003e\u003e polarised([0, 90])\\n\\n    ans = 0]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":59721,"title":"finding vector pair with min angle between them","description":"given a matrix with more than one row , compare row vectors of the given matrix and find the pair with the minumum angle between them , \"without using the dot fucntion\" \r\nyou can find the angle from the following formula\r\n θ = cos-1 [ (a. b) / (|a| |b|) ]\r\nthe product between the two vectors is the dot product \r\na⋅b=∑(ai​)*(bi)   from i=1 to n​\r\nthe length of a vector is the square root of the sum of the squares of the components\r\n","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 222px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 111px; transform-origin: 407px 111px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003egiven a matrix with more than one row , compare row vectors of the given matrix and find the pair with the minumum angle between them , \"without using the dot fucntion\" \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eyou can find the angle from the following formula\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e θ = cos-1 [ (a. b) / (|a| |b|) ]\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ethe product between the two vectors is the dot product \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003ea\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e⋅\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eb\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e∑(\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ea\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ei\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e​)*(\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eb\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ei)   from i=1 to n\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ethe length of a vector is the square root of the sum of the squares of the components\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [v1 v2] = minAngle(mat)\r\n   \r\nend","test_suite":"%%\r\nmat = [ 155    12   344   356\r\n   305   234   135    40\r\n   227    13   316   310\r\n   266    72    51   178\r\n     8   195   166   289];\r\nres1=[155    12   344   356];\r\nres2=[227    13   316   310];\r\n[v1 v2]=minAngle(mat);\r\nassert(isequal([v1 v2],[res1 res2]))\r\n\r\n%%\r\nmat=[  2    50    47\r\n    24    99    24\r\n    89    23    42\r\n    13    37     9\r\n    66     5    83\r\n     2     5    95\r\n    60    28     3\r\n    52    84    13\r\n    87    35    14\r\n    37    93    17\r\n    80    43   100\r\n    20    16    98\r\n    27    41     3\r\n    59    86    76\r\n    22    56    26\r\n    99    68    42\r\n    13    94    40];\r\n\r\n\r\n \r\n\r\n\r\n\r\n\r\n res1=[   13    37     9];\r\n  res2=[  37    93    17];\r\n [v1 v2]=minAngle(mat);\r\nassert(isequal([v1 v2],[res1 res2]))\r\n\r\n%%\r\nmat=[ 79     4    66    79    23\r\n    22    42    94     3    25\r\n    78    36    58    16    71\r\n    55    52    23    23     8\r\n    67     3     4    80    82\r\n    96    74    85    74    98];\r\n\r\n   res1=[ 78    36    58    16    71];\r\n\r\n\r\n\r\n   res2=[ 96    74    85    74    98];\r\n    [v1 v2]=minAngle(mat);\r\nassert(isequal([v1 v2],[res1 res2]))\r\n\r\n %%\r\n  mat=[ 8    47    50    69\r\n    78    49    11    83\r\n    68    53    63    98\r\n    97    74   100    57\r\n     4    48    24    91\r\n    96    46    12    92\r\n    64    61     4     1\r\n    73    61    72    71\r\n    78     6    47    63\r\n    52    34    18    41\r\n    46     3    95    38\r\n   100    10    52     2\r\n    88    71    59    25\r\n    63    33    99    43\r\n    95    56    48    44\r\n    73     6    13    14\r\n    56    60     8    84\r\n    96    60     7    85\r\n     8    90    54    76\r\n     1     6    39    70\r\n    83    12    18    44\r\n    44     9    85    11\r\n    65    43    17    90\r\n    44    68    36    34\r\n    92    89    46    89\r\n    36    37    65    79\r\n    11    69    76    41\r\n    41    17    23    56\r\n    94    10    16     7\r\n    51    98    17    70\r\n    94    18    53    11\r\n    45     9    38    42\r\n    59    99    32    31\r\n    81    85    12    82\r\n    46    48    87   100\r\n    52    17    78    17\r\n    35    55    61    57\r\n    46    55    22    93\r\n    52    51    31    10\r\n    84    56    31    34\r\n    77    51    62    66\r\n    41    88    17    40\r\n    90    62    86    20\r\n    86    50    60    48\r\n    11    50     3    54\r\n    31    52     5    71\r\n     1     3    98    47\r\n    28    86    93    55\r\n    71    55    16    93\r\n    84     1    68     7\r\n    17    64    98    33\r\n    76    19     2    72\r\n    72     1    56    73\r\n    13    92    67    14\r\n    49     9    92    37\r\n    44    28    83    76\r\n    56    78    58    20\r\n     6     3    75    90\r\n    96    21    63    10\r\n    59     2    60    72\r\n    17    65    28    98\r\n    34    46     5    41\r\n    48    56    54    47\r\n    76    95    90     8\r\n    57    35     5    54\r\n    26    30    13    73\r\n     6    49    17    46\r\n    23    59    82    56\r\n    55    46    46    33\r\n    80    15    42     6\r\n    78    57    59    22\r\n    96    11    69    23\r\n    72    58   100    42\r\n    88    48    23    23\r\n    16    35    48    32\r\n    11    45    79    61\r\n    95    82    31    63\r\n    19    49    73    19\r\n    65    14    81    41\r\n    12    81     4    30\r\n    70    24    33    82\r\n     7     5    66    14\r\n    98    79     3    37\r\n     9    81    91    72\r\n    50    62    99    11\r\n     9    66    18    58\r\n    35    38    29    61\r\n    36     9    44     5\r\n    98    25    92    84\r\n    53     9    33    18\r\n    85    80    93     6\r\n    60    56    25    10\r\n     9    29    90    32\r\n    36    98    38    84\r\n    26    64    21    21\r\n    52    36    58    48\r\n    92    15    35    18\r\n    62    37    98    67\r\n    79    53    24     9\r\n    12    62   100    65\r\n    30    26    24    61\r\n    84    12    29    17];\r\n     res1= [36    37    65    79];\r\n\r\n\r\n\r\n\r\n  res2=[ 46    48    87   100];\r\n  [v1 v2]=minAngle(mat);\r\nassert(isequal([v1 v2],[res1 res2]))\r\n%%\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":5,"created_by":3838707,"edited_by":3838707,"edited_at":"2024-03-28T08:31:50.000Z","deleted_by":null,"deleted_at":null,"solvers_count":8,"test_suite_updated_at":"2024-03-28T08:31:50.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2024-03-27T20:22:32.000Z","updated_at":"2026-01-06T16:39:24.000Z","published_at":"2024-03-27T20:23:40.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003egiven a matrix with more than one row , compare row vectors of the given matrix and find the pair with the minumum angle between them , \\\"without using the dot fucntion\\\" \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eyou can find the angle from the following formula\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e θ = cos-1 [ (a. b) / (|a| |b|) ]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ethe product between the two vectors is the dot product \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e⋅\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eb\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e=\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e∑(\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ei\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e​)*(\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eb\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ei)   from i=1 to n\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e​\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ethe length of a vector is the square root of the sum of the squares of the components\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":44372,"title":"Polarisation","description":"You have n polarising filters stacked one on top of another, and you know each axis angle. How much light gets passed through the filter bank? For more information, see \u003chttps://en.wikipedia.org/wiki/Polarizer Polarizer (Wikipedia)\u003e.\r\n\r\n    \u003e\u003e n = [0, 90];\r\n    \u003e\u003e polarised([0, 90])\r\n\r\n    ans = 0","description_html":"\u003cp\u003eYou have n polarising filters stacked one on top of another, and you know each axis angle. How much light gets passed through the filter bank? For more information, see \u003ca href = \"https://en.wikipedia.org/wiki/Polarizer\"\u003ePolarizer (Wikipedia)\u003c/a\u003e.\u003c/p\u003e\u003cpre\u003e    \u0026gt;\u0026gt; n = [0, 90];\r\n    \u0026gt;\u0026gt; polarised([0, 90])\u003c/pre\u003e\u003cpre\u003e    ans = 0\u003c/pre\u003e","function_template":"function y = polarised(x)\r\n  y = max(x);\r\nend","test_suite":"%%\r\nx = 5;\r\ny_correct = 0.5;\r\nassert(abs(polarised(x)-y_correct) \u003c 1e-10)\r\n%%\r\nx = 180;\r\ny_correct = 0.5;\r\nassert(abs(polarised(x)-y_correct) \u003c 1e-10)\r\n%%\r\nx = 365;\r\ny_correct = 0.5;\r\nassert(abs(polarised(x)-y_correct) \u003c 1e-10)\r\n%%\r\nx = [91, 1];\r\ny_correct = 0;\r\nassert(abs(polarised(x)-y_correct) \u003c 1e-10)\r\n%%\r\na = randi([-360, 360]);\r\nb = 90*(1+2*randi([-3, 3]));\r\nx = [a, a+b];\r\ny_correct = 0;\r\nassert(abs(polarised(x)-y_correct) \u003c 1e-10)\r\n%%\r\na = randi([-360, 360]);\r\nb = 90*(1+2*randi([-3, 3]));\r\nx = [a, a+b];\r\ny_correct = 0;\r\nassert(abs(polarised(x)-y_correct) \u003c 1e-10)\r\n%%\r\nx = [0, 22.5];\r\ny_correct = 0.85355339059/2;\r\nassert(abs(polarised(x)-y_correct) \u003c 1e-10)\r\n%%\r\nx = [0, -45];\r\ny_correct = 0.25;\r\nassert(abs(polarised(x)-y_correct) \u003c 1e-10)\r\n%%\r\nx = [5, 140];\r\ny_correct = 0.25;\r\nassert(abs(polarised(x)-y_correct) \u003c 1e-10)\r\n%%\r\nx = 5 + (1:5)*22.5;\r\ny_correct = 0.53079004294/2;\r\nassert(abs(polarised(x)-y_correct) \u003c 1e-10)\r\n\r\n\r\n\r\n","published":true,"deleted":false,"likes_count":4,"comments_count":10,"created_by":14358,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":269,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":34,"created_at":"2017-10-10T21:58:52.000Z","updated_at":"2026-03-26T15:36:11.000Z","published_at":"2017-10-16T01:45:10.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYou have n polarising filters stacked one on top of another, and you know each axis angle. How much light gets passed through the filter bank? For more information, see\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Polarizer\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePolarizer (Wikipedia)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[    \u003e\u003e n = [0, 90];\\n    \u003e\u003e polarised([0, 90])\\n\\n    ans = 0]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" 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