Grouping vectors into groups of 3 by attribute

Three groups are given, each with eight -vectors:

V1=
-0.0953681318008175	0.858850259468781	0.465327737835860
-0.743367248943597	0.315573550722436	-0.557734004373653
0.683550749866093	-0.246647785705382	0.659675673733952
-0.181058614839515	-0.767007049361869	-0.584950182984675
0.118761748181243	0.743380411572561	0.629706251526340
-0.721912423469330	0.520402704143404	-0.413850493189363
0.684870513276088	-0.444565219875274	0.544575887892960
0.00541628034354009	-0.874348540602108	-0.445795968809064

V2=
0.288328864090772	0.663613522812372	-0.690270583470916
-0.0661655475859850	0.714486376794470	0.696506466316257
0.0659856355945583	-0.714480201575689	-0.696529868264201
-0.290257865507787	-0.355082937133591	0.888626130158261
0.293720151348812	0.453108073152230	-0.841671828368103
0.0218190503132586	0.884590829792312	0.465846447687839
-0.0107163145532371	-0.867045341360451	-0.498103861217854
-0.293858044300728	-0.555374924488560	0.777943483153936

V3=
-0.983704829430046	0.127754733568269	-0.126193025298098
0.569942911640406	0.507765297260453	-0.645957137429290
-0.600866408147646	-0.493858673852356	0.628477774346492
0.913781057203716	-0.270305866504114	0.303084247456218
-0.902987180751040	0.284301799787109	-0.322036925000972
0.766156223036288	0.391842259005210	-0.509298958360849
-0.772797513782289	-0.387356431991189	0.502653101367087
0.984595406214300	-0.125219791950337	0.121708803266623

Task:

it is necessary to take a vector from each group and compose a basis in such a way that the angles between the corresponding pairs of vectors (i.e. ) are the same or differ insignificantly, i.e. the result should be 8 bases that meet these requirements.

How to do it in MATLAB?

[Matt J]

Three groups are given, each with eight 3x3 -vectors:

Don't you mean with three 8x3 matrices?

[Andrew Sol]

each row of a separate group is a vector.

[Matt J]

Your V matrices are 8x3 not 3x3.

[Andrew Sol]

Matt, each group, a collection of 8 vectors arranged in rows. Each line should be treated like a vector.