Question

A directed line segment makes angles 45° and 60° with x-axis and y-axis and an acute angle with z-axis. If P (– 1, 2, – 3) and Q (4, 3, 1) are two points in space, find the projection of PQ on the given line.

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Solution

Let l, m, n be the direction cosines of the given line AB. Then

straight l space equals space cos space 45 degree comma space space space equals space fraction numerator 1 over denominator square root of 2 end fraction comma space space straight m space equals space cos space 60 degree space equals space 1 half

Now

straight l squared plus straight m squared plus straight n squared space equals space 1 space space space space space rightwards double arrow space space space space space 1 half plus 1 fourth plus straight n squared space equals space 1 space space space rightwards double arrow space space space straight n squared space equals space 1 fourth

therefore space space space straight n space equals space 1 half

[∵ line AB makes acute angle with z-axis]

therefore

direction cosines of line AB are

fraction numerator 1 over denominator square root of 2 end fraction comma space 1 half comma space 1 half.

Now, P, Q are (-1, 2, -3), (4, 3, 1).

therefore

projection of PQ on AB = (4+1).

fraction numerator 1 over denominator square root of 2 end fraction plus left parenthesis 3 minus 2 right parenthesis. space 1 half plus left parenthesis 1 plus 3 right parenthesis. space 1 half

open square brackets because space space of space left parenthesis straight x subscript 2 minus straight x subscript 1 right parenthesis space straight l space plus space left parenthesis straight y subscript 2 minus straight y subscript 1 right parenthesis space straight m space plus left parenthesis straight z subscript 2 minus straight z subscript 1 right parenthesis space straight n close square brackets

equals space fraction numerator 5 over denominator square root of 2 end fraction plus 1 half plus 4 over 2 space equals space fraction numerator 5 over denominator square root of 2 end fraction plus 5 over 2 space equals space fraction numerator 5 square root of 2 plus 5 over denominator 2 end fraction space equals 5 over 2 left parenthesis square root of 2 plus 1 right parenthesis

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