Vector Multiplication Methods Explained

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 0.312 x vcos37 Bsin60 37 vsin37 4.00107 60 v x Bcos60

6. Sample Nasty Problem (cont.)