Let `A(-1, 2, 1) and B(4, 3, 5)` be two given points. Find the projection of AB on a line which makes angle `120^(@) and 135^(@)` with Yand Z-axes respectively, and an acute angle with 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