Name
MATH 253
Paper Homework 1
Sections 501-503
Fall 2008
P. Yasskin
Prove the Pythagorean Identity for Vectors:
⃗ ⋅ ⃗v  2 + |u
⃗ × ⃗v | 2 = |u
⃗ | 2 |v⃗| 2
u
as follows: Consider the vectors ⃗
u = u 1 , u 2 , u 3  and ⃗v = v 1 , v 2 , v 3 .
Compute each of the following by hand on paper. Show your work. Simplify where possible.
1. Write the algebraic definition for ⃗
u ⋅ ⃗v.
⃗
u ⋅ ⃗v =
⃗ ⋅ ⃗v 
2. Write out u
2
to get 6 terms.
⃗ ⋅ ⃗v  2 =
u
3. Write the algebraic definition for ⃗
u × ⃗v.
⃗
u × ⃗v =
⃗ × ⃗v |
4. Write out |u
2
to get 9 terms.
⃗ × ⃗v | 2 =
|u
⃗ ⋅ ⃗v  + u
⃗ × ⃗v 
5. Add u
2
2
and cancel some terms.
⃗ ⋅ ⃗v  2 + |u
⃗ × ⃗v | 2 =
u
⃗ | |v⃗| .
6. Multiply out |u
2
2
⃗ | 2 |v⃗| 2 =
|u
7. Are the answers to (5) and (6) equal?