ba

axb

θ

Properties of the Cross Product

1. axb = -bxa

2. (ca)xb = c(axb) = ax(cb)

3. ax(b+c) = axb + axc

4. (a+b)xc = axc + bxc

5. a.(bxc) = (axb).c

6. ax(bxc) = (a.c)b-(a.b)c

The so-called scalar triplea.(bxc)

122131132332

122131132332

, ,)(

, ,

cbcbcbcbcbcbacba

cbcbcbcbcbcbcb

But recall that the components of (bxc) come from 2x2 determinants

a.(bxc)

321

321

321

32

323

31

312

32

321

)(

)(

ccc

bbb

aaa

cba

cc

bba

cc

bba

cc

bbacba

A-ha! We have a quick way to compute this!

The Geometric InterpretationVolume of a parallelpiped

h

b

a

c

bxc

Volume = (Area of Base)*(height)

V = |bxc| |a||cos( )|=|a.(bxc)|

Q. How can we use the cross

product to determine if 3 vectors are

coplanar (lie in the same plane)?

A.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)