Let ` vec u , vec va n d vec w` be such that `| vec u|=1,| vec v|=2a n d| vec w|=3.` If the projection of ` vec v` along ` vec u` is equal to that of ` vec w` along ` vec u` and vectors ` vec va n d vec w` are perpendicular to each other, then `| vec u- vec v+ vec w|` equals `2` b. `sqrt(7)` c. `sqrt(14)` d. `14`

Options

A. 2
B. `sqrt7`
C. `sqrt(14)`
D. 14

Answer

C

Explanation

Since, `|u| = 1, |v| = 2,|w| = 3`
The projection of v along `u = (v.u)/(|u|)` and the projection of w along `u = (w.u)/(|u|)`.
According to given condition.
`(v.u)/(|u|) = (w.u)/(|u|)`
`rArr v.u = w.u "…."(i)`
Since, v and w are perpendicular to each other.
`:. v. w = 0`
Now, `| u- v + w|^(2) = |u|^(2) + |v|^(2) +|w|^(2)`
`- 2u . v - 2v.w+2u.w`
`rArr |u- b- v + w|^(2) = 1 + 4+ 9 - 2 u . v + 0 + 2u . v` [from Eq. (i) ]
`rArr |u - v + w|^(2) = 1 + 4 + 9`
`:. |u- v + w| = sqrt(14)`