1. Dot Products There are two ways to multiply two vectors
The dot product produces a scalar quantity
It has no direction
It can be pretty easily computed from geometry
It can be easily computed from components
The dot product of two unit vectors is easy to memorize
The dot product is commutative

2. Cross Products The cross product produces a vector quantity
It is perpendicular to both vectors
Requires the right-hand rule
Its magnitude can be easily computed from geometry
It is a bit of a pain to compute from components

3. Determinants
Finding the cross product requires that you memorize the formula, or know how to compute determinants
Computing a 33 determinant:
Multiply on the diagonal down-right
Add the other two down-rights, wrapping as needed
Subtract the diagonal down-left
Subtract the other two down-lefts, wrapping as needed
Simplify

4. Simple Rules for Cross-Products
Vectors that are parallel or anti-parallel have zero cross product b a
Cross products are anti-symmetric c
Basis vectors:
Any vector with itself gives zero
Think of ijk as a circle: any two in order gives the third
Any two in reverse order gives minus the third

5. Sample Nasty Problem
An electron moving in the xy-plane at a speed of 4.00 m/s at an angle of 37 below the x-axis enters a region where the magnetic field is 312 mT in the xz-plane and pointed at a 60 angle above the x-axis. What is the acceleration of the electron? B y z x
0.312
vcos37
37
Bsin60
4.00107
vsin37
60 v x
Bcos60

6. Sample Nasty Problem (cont.)