11.4 The Cross productFor an animation of this topic visit:

http://www.math.umn.edu/~nykamp/m2374/readings/crossprod/

Red x Green = Blue

What is the cross product?• a × b is a vector that is perpendicular to both a

and b.

• ||a × b|| is the area of the parallelogram spanned by a and b (i.e. the parallelogram whose adjacent sides are the vectors a and b).

• The direction of a×b is determined by the right-hand rule. (This means that if we curl the fingers of the right hand from a to b, then the thumb points in the direction of a × b.)

Definition of Cross Product of Two Vectors in Space

Find the cross product of vectors (by hand and with a calculator)

u = 3i +2j – kv = i + 2kNote: To find a cross product on the TI89Press 2nd 5 (math) – 4 matrix – L Vector ops - crossPcrossP([3,2,-1],[1,0,2])

Definition of Cross Product of Two Vectors in Space

For an explanation and animation of the cross product visit:

http://www.math.umn.edu/~nykamp/m2374/readings/crossprod/

Example 1

• Find the cross product of the two vectors

• u = i – 2j + k v = 3i + j – 2k

• Find u x v

• Find v x u

• Find v x v

Example 1

• Find the cross product of the two vectors

• u = i – 2j + k v = 3i + j – 2k

• Find u x v

• Find v x u

• Find v x v

Example 1 continued

• u = i – 2j + k v = 3i + j – 2k

• Find a unit vector that is orthogonal to u and v

The Triple Scalar Product

Geometric Property of Triple Scalar

Example 5

• u = 3i – 5j + k• v = 2j – 2k• w = 3i + j + k• Find the volume of

the parallelepiped having vectors u, v and w for sides

Example 5 hint

This works because bxc yields a vector perpendicular to a and b in with the magnitude of the area of the parallelogram formed (the base of the parallel piped)

bxc points in the direction of the height of the parallelepiped when dotted with a this gives the magnitude of bxc times the portion of a that points in the direction of bxc (the direction of the height )in other words this gives us the same as the formula Bh

Q: Why should you never make a math teacher angry?

A: You might get a cross product

Q: What do you get when you cross an elephant and a banana? A: | elephant | * | banana | * sin(theta) 

Proof of the cross productproof that the cross product is orthogonal to the two original

vectors is part of the homework

│u x v │ = │u │ │v │sinθ