Multivariate Calculus - Bruce E. Shapiro

18 LECTURE 2. VECTORS IN 3D
Figure 2.6: A vector u is expressed as the sum of its components m parallel to u
and n perpendictular to u.
Definition 2.11 The projection of a vector v on u is
pr u ((v)) = m = (v · û) û (2.34)
Example 2.7 Express v = (−3, 2, 1) as the sum of vectors m parallel and n perpendicular
to u = (−3, 5, −3).
Solution.
‖u‖ = √ (−3) 2 + (5) 2 + (−3) 2 = √ 43
û = u
‖u‖ = √ 1 (−3, 5, −3)
43
1
v · û = √ ((−3)(−3) + (5)(2) + (−3)(1)) = √ 16
43 43
m = (v · û) û = 16
(
43 (−3, 5, −3) = − 48
43 , 80 )
43 , −48 43
(
n = v − m = (−3, 2, 1) − − 48
43 , 80 )
43 , −48 = 1 (3, −26, 31) .
43 7
Example 2.8 Express v = (2, −1, −2) as the sum of vectors m parallel and n
perpendicular to u = (2, 4, 5).
Solution.
‖u‖ = √ (2) 2 + (4) 2 + (5) 2 = √ 45
û = u
‖u‖ = √ 1 (2, 4, 5) 45
1
v · û = √ ((2)(2) + (−1)(4) + (−2)(5)) = −√ 10
45 45
m = (v · û) û = − 10
45 (2, 4, 5) = −1 (4, 8, 10)
9
n = v − m = (2, −1, −2) − 1 (
9 (4, 8, 10) = − 81
43 , 6 43 , 91 )
43
Revised December 6, 2006. Math 250, Fall 2006