Simplifying Vector Expressions Involving Dot and Cross or Vector Products

When simplifiying expression involving the dot and/or cross product of vectors the following identities are useful.

(1)

(2)

(3)

and (4)

and(5)

Example: Suppose we want to simplify

Identity (5) gives

Then using the second rule (5) gives

Obviouslyand

Rule (3) givesandboth equal zero so the expression simplifies to

Now use rule (2) to giveThe expression becomes

Example: Simplify

Use rule (4) to give

Now use (4) again, twice, to give

Obviouslyand

The expression becomes

Now use rule (1) to write

The expression becomes