Cross Products

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January 27th 2007, 12:46 PM
Ideasman

Cross Products

Anyone have an easy way for computing cross-products? I came across:

a x b = A_(x)b = [[0, -a_3, a_2],[a_3, 0, -a_1],[-a_2, a_1, 0]] multiplied with vector [b_1, b_2, b_3].

which seems very useful.

My book attempts to solve it by using determinants;

Also, if we have two vectors, a and b, in R^2, is a x b defined?

I've only seen this done with vectors in R^3.

Lastly, if we have two vectors a and b, I'm assuming those vectors are related to the a x b where their product is = 0, thus they're orthogonal?

Thanks.
January 27th 2007, 12:59 PM
Glaysher

Quote:

Originally Posted by Ideasman

Also, if we have two vectors, a and b, in R^2, is a x b defined?

Thanks.

I can't see how as the defintion of cross product involves the vector n which is perpendicular to both vectors that you are finding the product of. This will ususally be impossible just working in the x-y plane
January 27th 2007, 02:14 PM
ThePerfectHacker

Quote:

Originally Posted by Ideasman

.

Lastly, if we have two vectors a and b, I'm assuming those vectors are related to the a x b where their product is = 0, thus they're orthogonal?
.

No the cross product of two non-zero vectors is parallel if and only if they are parallel.
January 27th 2007, 02:33 PM
Soroban

Hello, Ideasman!

Quote:

Anyone have an easy way for computing cross-products?
I came across:

$a \times b \;=\;\begin{bmatrix}0 & -a_3 & a_2 \\ a_3 & 0 &-a_1 \\ -a_2 & a_1& 0\end{bmatrix}\,\begin{bmatrix}b_1 \\ b_2 \\ b_3\end{bmatrix}$ . which seems very useful.

My book attempts to solve it by using determinants.

I prefer the determinant method.
I already know how to evaluate a 3-by-3 determinant.
. . Why would I want to memorize this new (and very strange) formula?

Quote:

Also, if we have two vectors, $\vec{a}$ and $\vec{b}$ in $R^2$ , is $\vec{a} \times \vec{b}$ defined?

Not in $R^2$ .

$\vec{a} \times \vec{b}$ is perpendicular to both $\vec{a}$ and $\vec{b}$ . .It is automatically in $R^3$ .

Quote:

If we have two vectors $\vec{a}$ and $\vec{b}$ ,
I'm assuming those vectors are related to the $\vec{a} \times \vec{b}$
where their product is = 0, thus they're orthogonal?

Sorry, this makes no sense.

Are these are any two vectors?
. . Then you mention their cross product . . . why?

"Their product is zero" . . . What product? .Dot product? .Cross product?

If their cross product is zero, the vectors are parallel.
If their dot product is zero, the vectors are orthogonal.

So what exactly is your question?