Examples for The projection of a vector

Example 1

Given v = i - 2j + 2k and u = 4i - 3k find

We have u · v = -2, |v| = 3 and |u| = 5.

The component of v in the direction of u is $u-· v --2 |u| = 5 .$
The projection of v in the direction of u is $u-· v --2 8-- 6-- |u |2 u = 52 (4i- 3k) = - 25i + 25k.$
To calculate the resolution of v into components parallel and perpendicular to u we find the component parallel to u and that is just the projection of v in the direction of u - this we have calculated in part (b). The component perpendicular to u is $u · v 8 6 33 44 v - ----2 u = i- 2j + 2k - (- --i + --k) = --i- 2j + --k. |u| 25 25 25 25$