Simplifying Vector Expressions Involving Dot and Cross or Vector Products
When simplifying 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
Obviously
and
Rule (3) gives
and
both equal zero so the expression simplifies to
Now use rule (2) to give
The expression becomes
Example: Simplify
Use rule (4) to give
Now use (4) again, twice, to give
Obviously
and
The expression becomes
Now use rule (1) to write
The expression becomes