Question

If A and B are the points (– 1, 2, 1) and (4, 3, 5). Find the projection of AB on a line which makes angles of 120° and 135° with y-axis and z-axis respectively and acute angle with x-axis.

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Solution

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

straight m equals space cos space 120 degree space equals space cos space left parenthesis 180 degree space minus space 160 degree right parenthesis space equals space minus cos space 60 degree space equals space minus 1 half. straight n space equals space cos space 135 space degree space equals space cos space left parenthesis 180 degree minus 45 degree right parenthesis space equals space minus cos space 45 degree space equals negative fraction numerator 1 over denominator square root of 2 end fraction

Now,

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

therefore space space space straight l space equals space 1 half

[∵ line PQ makes acute angle with x-axis]

therefore

direction cosines of PQ are

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

Now A and B are (–1, 2, 1), (4, 3, 5) respectively

therefore

projection of AB on PQ =

left parenthesis 4 plus 1 right parenthesis. space open parentheses 1 half close parentheses plus space left parenthesis 3 minus 2 right parenthesis space open parentheses negative 1 half close parentheses plus left parenthesis 5 minus 1 right parenthesis. space open parentheses negative fraction numerator 1 over denominator square root of 2 end fraction close parentheses

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 left parenthesis straight y subscript 2 minus straight y subscript 1 right parenthesis space straight m space plus space left parenthesis straight z subscript 2 minus straight z subscript 1 right parenthesis space straight n close square brackets

equals negative 5 over 2 minus 1 half minus fraction numerator 4 over denominator square root of 2 end fraction space equals space 2 minus square root of 2

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