Question

Given vector u = (0, 3, 3) and vector v = (-1, 1, 1).
a) Find the projection of u onto v, and
b) Find the vector component of u orthogonal to v.
[Give your answers in the form of linear combination of the standard unit vectors (i , j , and k ) ]

Answer 1

Question #17344Given vector $u = (0,3,3)$ and vector $v = (-1,1,1)$. a) Find the projection of $u$ onto $v$, and b) Find the vector component of $u$ orthogonal to $v$. [Give your answers in the form of linear combination of the standard unit vectors $(i,j,\text{and} k)$].

Solution. a) projection of $u$ onto $v$ is given by the formula $proj_v(u) = \frac{(u,v)}{(v,v)} v = \frac{6}{3} v = 2v$, so $proj_v(u) = -2i + 2j + 2k$.

b) This component equals $u - proj_v(u) = (2,1,1) = 2i + j + k$.

Answer.a) $-2i + 2j + 2k$, b) $2i + j + k$.