The Right Hand Rule b a axb.

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Presentation transcript:

The Right Hand Rule b a axb

Properties of the Cross Product axb = -bxa (ca)xb = c(axb) = ax(cb) ax(b+c) = axb + axc (a+b)xc = axc + bxc a.(bxc) = (axb).c ax(bxc) = (a.c)b-(a.b)c

The so-called scalar triple a.(bxc) But recall that the components of (bxc) come from 2x2 determinants

a.(bxc) A-ha! We have a quick way to compute this!

The Geometric Interpretation Volume of a parallelpiped V = |bxc| |a||cos( )|=|a.(bxc)| bxc a h c b Volume = (Area of Base)*(height)

Q. How can we use the cross product to determine if 3 vectors are coplanar (lie in the same plane)? Determine if the volume of the resulting parallelpiped is nonzero

Example Do the following points lie in the same plane? A=(1,-1,2) B=(2,0,1) C=(3,2,0) D=(5,4,2)