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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.
Solution
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"`.
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