Question

Find a vector of magnitude 11 in the direction opposite to that of `vec"PQ"` where P and Q are the points (1, 3, 2) and (–1, 0, 8), respectively.

Answer 1

The vector with initial point P (1, 3, 2) and terminal point Q (–1, 0, 8) is given by

`vec"PQ" = (-1 - 1) hat"i" + (0 - 3) hat"j" + (8 - 2) hat"k"`

= `-2hat"i" - 3hat"j" + 6hat"k"`

Thus `vec"OP" = - vec"PQ" = 2hat"i" + 3hat"j" - 6hat"k"`

⇒ `|vec"OP"| = sqrt(2^2 + 3^2 + (-6)^2)`

= `sqrt(4 + 9 + 36)`

= `sqrt(49)`

= 7

Therefore, unit vector in the direction of `vec"OP"` is given by

`hat"OP" = vec"OP"/|vec"OP"|`

= `(2hat"i" + 3hat"j" - 6hat"k")/7`

Hence, the required vector of magnitude 11 in direction of `vec"OP"` is 11

`hat"OP" = 11((2hat"i" + 3hat"j" - 6hat"k")/7)`

= `22/7hat"i" + 33/7hat"j" - 66/7 hat"k"`.