Find the length of the projection of the line segment joining the points P (3, -1, 2) and Q(2, 4, -1) on the line with direction ratios -1, 2, -2.

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Solution

Direction ratios of AB are –1, 2, –2.
∴ direction cosines of AB are

fraction numerator negative 1 over denominator square root of 1 plus 4 plus 4 end root end fraction comma space space fraction numerator 2 over denominator square root of 1 plus 4 plus 4 end root end fraction comma space fraction numerator negative 2 over denominator square root of 1 plus 4 plus 4 end root end fraction

i.e.,

negative 1 third comma space 2 over 3 comma space minus 2 over 3

P and Q are (3, -1, 2) and (2, 4, -1)

therefore

projection of PQ on AB

equals space left parenthesis 2 minus 3 right parenthesis space open parentheses negative 1 third close parentheses plus left parenthesis 4 plus 1 right parenthesis space open parentheses 2 over 3 close parentheses plus left parenthesis negative 1 minus 2 right parenthesis space open parentheses negative 2 over 3 close parentheses equals space 1 third plus 10 over 3 plus 6 over 3 space equals 17 over 3

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