Cross Product of Vectors

**collegemath** · February 5th 2011, 07:38 PM

I need help answering this question.

"Use vectors v=2i+3j-k and z+-3i-j-4k to prove that -(v X z) = z X v. Write a summary and show graphically what this means with respect to the cross product v X z."

I get 13i + 11j + 7k as the cross product of both -(v X z) and z X v so I understand how to prove this algebraically, but I have no idea how i would write a summary about this or show this graphically.

Help is greatly appreciated.

**dwsmith** · February 5th 2011, 07:48 PM

Originally Posted by collegemath

I need help answering this question.

"Use vectors v=2i+3j-k and z+-3i-j-4k to prove that -(v X z) = z X v. Write a summary and show graphically what this means with respect to the cross product v X z."

I get 13i + 11j + 7k as the cross product of both -(v X z) and z X v so I understand how to prove this algebraically, but I have no idea how i would write a summary about this or show this graphically.

Help is greatly appreciated.

$\displaystyle -(v\times z)=(-1)\begin{vmatrix}i&j&k\\2&3&-1\\-3&-1&-4\end{vmatrix}=-[i(-12-1)-j(-8-3)+k(-2+9)]$

$\displaystyle (z\times v)=\begin{vmatrix}i&j&k\\-3&-1&-4\\2&3&-1\end{vmatrix}=i(1+12)-j(3+8)+k(-9+2)$

**collegemath** · February 5th 2011, 07:58 PM

so how would i write a summary on this or show this graphically?

**dwsmith** · February 5th 2011, 08:00 PM

Originally Posted by collegemath

so how would i write a summary on this or show this graphically?

Draw the vectors and the corresponding normal vector (graphically). Since it says write or show graphically, I would show graphically.

**CaptainBlack** · February 5th 2011, 08:02 PM

Originally Posted by dwsmith

$\displaystyle -(vxz)=(-1)\begin{vmatrix}i&j&k\\2&3&-1\\-3&-1&-4\end{vmatrix}=-[i(-12-1)-j(-8-3)+k(-2+9)]$

$\displaystyle (zxv)=\begin{vmatrix}i&j&k\\-3&-1&-4\\2&3&-1\end{vmatrix}=i(1+12)-j(3+8)+k(-9+2)$

LaTeX

[tex] z\times v[/tex]

gives:

$z\times v$

**collegemath** · February 5th 2011, 08:07 PM

which vectors do i draw and what is a correspnoding normal vector?

**dwsmith** · February 5th 2011, 08:09 PM

Originally Posted by collegemath

which vectors do i draw and what is a correspnoding normal vector?

Draw vectors v, z, and z x v is the normal vector.

Then draw -v, -z, and -(v x z) is the normal vector.

**collegemath** · February 5th 2011, 08:22 PM

looks like i also need to write a summary how do i do that?

**dwsmith** · February 5th 2011, 08:23 PM

Originally Posted by collegemath

looks like i also need to write a summary how do i do that?

I would first draw the plane v and z and the plane -v and -z. What do you notice?

**collegemath** · February 5th 2011, 08:38 PM

so is z X v just a reflection above the x - axis of v X z

**dwsmith** · February 5th 2011, 08:49 PM

Originally Posted by collegemath

so is z X v just a reflection above the x - axis of v X z

I am really bad at drawing in 3d. I would say it is a reflection across the line perpendicular to the vector.

I saw this by looking at a 2d example.

vector <1,0> <-1,0> this would be a reflection across the y axis which is perpendicular to it.

So I believe you could say is that -(v x z) is a reflection across the normal vector of (z x v).

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