P and Q are the points `(-1, 2, 1) and (4, 3, 5)` respectively. Find the projection of PQ on a line which makes angles of `120^@ and 135^@` with y and z-axes respectively and an acute angle with the x-axis.

Explanation

Let ` alpha ` be an acute angle that the given line make with x-axis. Then
`cos^(2) alpha= 1- (1)/(4) - (1)/(2) = (4 -2 -1)/(4) = (1)/(4)`
`rArr cos alpha = pm (1)/(2) " but " alpha` is acute
`therefore cos alpha = + ve `
`rArr cos alpha = (1)/(2) = cos 60^(@)`
`rArr alpha = 60^(@)`
Thus the diraction cosines of the given straight line are `cos 60^(@) , cos 120^(@) , cos135^(@)`
i.e., `(1)/(2), - (1)/(2), - (1)/(sqrt(2))`
Hence the projection of AB on the line
`(1)/(2)(4+1)- (1)/(2)(3-2) - (1)/(sqrt(2))(5-1) = (5)/(2) - (1)/(2) - 2sqrt(2)`
`= (2 -2sqrt(2))` units