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# Calculate the volume of the area enclosed by the following vectors.

Asked by Connor on 12 Feb 2013

Hello!

I'm certainly a beginner when it comes to MatLAB, but I can find my way around eventually; except with this problem. I'm given the following to find, with the givens:

Use the determinate, or norm, to calculate the volume of the area enclosed by the following vectors.

```a = 2*i+1*j+1*k
```
```b= 2*i+-2*j+6*k
```
```c=0*i+0*j+9*k
```

This is all that's given to me. When I enter this into MatLAB, it returns the error message, "??? Undefined function or variable 'k'." When I first looked at this problem, my first thought was, what are "i," "j," and "k." But I came to the conclusion that "i," and "j," are just the imaginary units, but I still have no idea what "k" is. Any help is appreciated!

Thanks!

Connor on 12 Feb 2013

So how should the code look then? MatLAB says there's an error for "k."

Sven on 12 Feb 2013

You need to understand in your homework what i, j, and k actually mean. They are not variables for you to plug into an equation. What if your question stated:

```a = 2*x+1*y+1*z
```

What do x, y and z represent?

If you can answer that question, you'll be able to do this homework without any problems.

Connor on 12 Feb 2013

That's what I'm confused about. All that's given is what I've included, and our professor hasn't explained this. I've gone through all of the lecture slides that have been shown in class.

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Answer by Youssef KHMOU on 13 Feb 2013

hi, like other users said , you need to understand that i,j,and k represent the Unitary vectors for Orthonormal basis -> -> -> ( i, j, k) with coordinates : i(1,0,0), j(0,1,0) and k(0,0,1),

You can create these vectors right? then you can create also the vectors a,b,c , so what type of product that gives a volume ?

THINK OF : Mixed product, Determinant, Base , columns .

Connor on 13 Feb 2013

Thank you. I really appreciate it. I finally figured it out with the "i(1,0,0), j(0,1,0) and k(0,0,1)" part. I ended up getting a negative number, so I just changed it to a positive number and I got the answer correct. Could you possibly explain to me why it came out negative?

Youssef KHMOU on 13 Feb 2013

ok good, we can explain this Geometrically :

you take the cube formed by (i,j,k) as reference , obviously the volume is 1 . so negative value is an issue of Orientation :

If you make a "parallelepiped" from random 3d vectors , and it takes at least one symmetry to go back to the unitary volume built by the unit vectors then the Sign is < 0.

Example :

a volume obtained from a=-5x-7y-6z, b=-11x-8y-41z, c=-14x-77y-36z

The sign is positive V=29048, because you only need deformation to get a unitary volume , but no symmetry, is it clear ?