5. The cross product

The cross product or vector product of v and w, denoted v$\times$   w, is defined only in R$^3$. Geometrically, v$\times$   w is a vector with direction perpendicular to v and w, of length $\vert$v$\vert \vert$w$\vert \vert\sin \theta\vert$ (the area of the parallelogram with sides v and w). So far, this description fits two vectors (if the length is nonzero), one the negative of the other. If vectors are drawn from the origin, v$\times$   w is the one from whose end w appears to be counterclockwise from v. To remember this relationship, just remember that i$\times$   j$=$   k, and relate that picture to your v and w. Algebraically,

v$\times$   w$= \left ( {\det \left [\begin{array}{cc} v _ 2&v _ 3\\ w _ 2&w _ 3 \end{arra... ...\begin{array}{cc} v _ 1& v _ 2\\ w _ 1& w _ 2 \end{array} \right ] } \right )$,

(or you can write the middle term as $\det \left [\begin{array}{cc} v _ 3& v _ 1\\ w _ 3& w _ 1 \end{array} \right ]$, so that the three coordinates change cyclically without a minus sign). There is an easy way of remembering these relations: Use v$\times$   w$= \det \left [\begin{array}{ccc} \mbox{\bf i}&\mbox{\bf j}&\mbox{\bf k}\\ v _ 1& v _ 2& v _ 3\\ w _ 1&w _ 2&w _ 3 \end{array} \right ]$ and expand by cofactors of the first row. This doesn't make very good sense mathematically, as you usually don't take determinants of matrices with vectors as entries, but it is handy as an aid to remembering.

Cross products have several uses in this course:

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They are just what is needed to find a vector perpendicular to two given vectors in R$^3$.

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The angle $\theta$ between two vectors v$,$w in R$^3$ satisfies $\vert \sin \theta \vert = \frac {\vert \mbox{\bf v} \times \mbox{\bf w} \vert} {\vert \mbox{\bf v} \vert \vert \mbox{\bf w} \vert}$.

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For vectors in R$^2$, let us denote by $D($v$,$w$)$ the useful determinant

$D($v$,$w$) = \det \left [\begin{array}{cc} v _ 1& v _ 2\\ w _ 1& w _ 2 \end{array} \right ]$.

If you make v, w into vectors in R$^3$ by giving each a third component of 0, their cross product becomes $(0,0, D($v$,$w$) )$. Thus in R$^3$, it is easy to compute the cross product of two vectors in the $x,y$-plane, and the result lies along the $z$-axis.

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Two nonzero vectors have the same direction if and only if their cross product is 0. (If they are drawn from the origin, this would mean they lie along the same line.)

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In R$^2$, the area of the parallelogram with sides given by v and w is $A = \vert D($v$,$w$) \vert$.

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In (), the cross product of v, w in R$^2$ (or better, in the $x,y$-plane of R$^3$) points up in R$^3$ if w is counterclockwise from v and down if w is clockwise from v. Thus in R$^2$, w is counterclockwise from v precisely when $D($v$,$w$) > 0$.

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In R$^2$, the sine of the angle $\theta$ between two vectors is given by

$\sin \theta = \frac {D(\mbox{\bf v},\mbox{\bf w})} {\vert \mbox{\bf v} \Vert \mbox{\bf w} \vert}$.

$\sin \theta$ does distinguish between positive and negative $\theta$, but notice that $\sin 80 ^\circ = \sin 100 ^\circ$, for example, so $\sin \theta$ is not enough information to determine $\theta$ uniquely.