12.4 The Cross Product

Let n to be a unit normal (perpendicular)vector to the plane satisfying the right hand rule.

The cross product of the vectors u and vDefinition

where is the angle between u and v.

Note: If

For nonzero vectors u and v , we have:

u is parallel to v if and only if | u x v | = 0

Properties of the Cross Product (pg. 700) Let u, v, w be vectorsand let r, s be scalars, then:

1. (r u ) x (s v ) = rs (u x v)

2. u x (v + w ) = (u x v)+ u x w

3. v x u = - (u x v)

4. (u + w ) x u = (v x u ) x (w x u)

5. 0 x u = 0

6. u x (v x w) = (u w) v - (u v) w

* u x u =

Example. Find u x v if u = 2 i + k and v = -4 i + 3 j

Geometric Interpretation

Consider the parallelogram generated by u and v.

Area of the parallelogram = (height)(base)

height base

A formula for u x vIf u = u i + u j + u k and v = v i + v j + v kthen u x v = ( u i + u j + u k ) x (v i + v j + v k )

(Multiply out)

Using a Determinant

Example. u = 2 i + k v = -4 i + 3 j

Example. Find a unit vector perpendicular to the plane containingthe points P(1,-1,0), Q(2,1,-1) and R(-1,1,2)

Example. Find the area of the triangle with vertices P(1,-1,0)Q(2,1,-1) and R(-1,1,2)

TorqueWhen turning a bolt by applying a force Fto a wrench, we produce a torque thatcauses the bolt to rotate.

Example The magnitude of the torque generated by the force

F at the pivot point P in the figure is: