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

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

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

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

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

Sample Nasty Problem (cont.)