Find the projection of the line segment joining the points (1, 2, 3), (4, 3, 1) on the line with direction ratios 3, –6, –2.

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Solution

Let P (1, 2, 3), Q (4, 3, 1) be given points and AB be the line whose direction ratios are 3; – 6, 2.
Direction cosines of AB are

fraction numerator 3 over denominator square root of 9 plus 36 plus 4 end root end fraction comma space space fraction numerator negative 6 over denominator square root of 9 plus 36 plus 4 end root end fraction comma space space fraction numerator negative 2 over denominator square root of 9 plus 36 plus 4 end root end fraction space space straight i. straight e. comma space space 3 over 7 comma space minus 6 over 7 comma space minus 2 over 7

Projection of PQ on AB = (4-1) +

open parentheses 3 over 7 close parentheses plus space left parenthesis 3 minus 2 right parenthesis space open parentheses negative 6 over 7 close parentheses space plus space left parenthesis 1 minus 3 right parenthesis space open parentheses negative 2 over 7 close parentheses

equals space 9 over 7 minus 6 over 7 plus 4 over 7 space equals space 7 over 1 space equals space 1.

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