1. Cross Product of Vectors

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.

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

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

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

4. 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.

5. 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

$$z\times v$$

gives:

$z\times v$

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

7. 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.

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

9. 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?

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

11. 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).