# cross product

• Feb 18th 2013, 12:22 PM
Hartlw
cross product
In case you’re not familiar with it, the cross product is very handy in 3d geometry:

aXb is a vector perpendicular to a and b in direction given by right hand rule (rotate (screw) a into b thru the acute angle θ between them), and magnitude absinθ.

aXb = “det” |i, j, k ; a1, a2, a3 ; b1, b2, b3| , = |row; row; row|, (not obvious, proof in textbooks)

Ex (0,0,1) X (a1,a2,a3) = |i,j,k ; 0,0,1 ; a1,a2,a3 | = -a2i + a1j = (-a2,a1,0)

It follows that a.bXc = det |a;b;c|

i,j,k is orthonormal basis (also called e1,e2,e3). "det" not really determinant- write out as determinant and evaluate as determinant.

It would be visually clearer if I could print determinants, sorry.
• Feb 18th 2013, 12:46 PM
jakncoke
Re: cross product
Is you question to show that $\displaystyle a(b \times c) = Det(a;b;c)$ or are you telling us an interesting fact about cross products?
• Feb 18th 2013, 01:00 PM
ILikeSerena
Re: cross product
Quote:

Originally Posted by Hartlw
It would be visually clearer if I could print determinants, sorry.

You mean like this?

$\displaystyle \mathbf a \times \mathbf b = \det \begin{pmatrix}\mathbf i & \mathbf j & \mathbf k \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{pmatrix}$

or like this?

$\displaystyle \mathbf a \times \mathbf b = \begin{vmatrix}\mathbf i & \mathbf j & \mathbf k \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$
• Feb 18th 2013, 02:49 PM
Hartlw
Re: cross product
ILikeSerena:

In either case it’s the determinant, written with any convention you like.

aXb = i(a2b3-a1b2) –j(a1b3-b1a3) +k(a1b2-a1b2).

It’s just short hand for (a1i + a2j + a3k) X (b1i + b2j + b3k) and noting iXi =0, iXj = k, etc. Sketch a little right hand triad of i,j,k to instantly get all the possibilities.

Geometrically a.bXc is volume of the triad a,b,c, and is usually evaluated as det |a; b; c|

This isn’t a question, just a point of information, since I get the impression the cross product is not generally handy.
For a complete lucid explanation of cross product google “cross product” wiki.