Question

A, B, C, D are the points (1, – 1, 0), (2, 1, – 1), (– 3, 2, 2) and (0, – 2, – 1) respectively. Find the projection of AB on CD.

Solution

The given points are A (1, – 1, 0), B (2, 1, – 1), C (– 3, 2, 2) and D (0, – 2, – 1).
Direction ratios of CD are 0 + 3, – 2 – 2, – 1 – 2 i.e.. 3, – 4, 3
∴ Direction ratios of CD are 0 + 3, – 2 – 2, – 1 – 2 i.e.. 3, – 4, 3
∴ direction cosines of CD are
$fraction numerator 3 over denominator square root of 9 plus 16 plus 9 end root end fraction comma space fraction numerator negative 4 over denominator square root of 9 plus 16 plus 9 end root end fraction comma space fraction numerator negative 3 over denominator square root of 9 plus 16 plus 9 end root end fraction space space straight i. straight e. comma space space space fraction numerator 3 over denominator square root of 34 end fraction comma space minus fraction numerator 4 over denominator square root of 34 end fraction comma space minus fraction numerator 3 over denominator square root of 34 end fraction$
$therefore space space space projection space of space AB space on space CD space equals space left parenthesis 2 minus 1 right parenthesis space open parentheses fraction numerator 3 over denominator square root of 34 end fraction close parentheses plus left parenthesis 1 plus 1 right parenthesis space open parentheses fraction numerator negative 4 over denominator square root of 34 end fraction close parentheses plus left parenthesis 1 minus 0 right parenthesis space open parentheses negative fraction numerator 3 over denominator square root of 34 end fraction close parentheses$
$open square brackets because space space of space space left parenthesis straight x subscript 2 minus straight x subscript 1 right parenthesis space straight l space plus space left parenthesis straight y subscript 2 minus straight y subscript 1 right parenthesis space straight m space plus space left parenthesis straight z subscript 2 minus straight z subscript 1 right parenthesis space straight n close square brackets$
(in magnitude)
$equals space fraction numerator 3 over denominator square root of 34 end fraction minus fraction numerator 8 over denominator square root of 34 end fraction plus fraction numerator 3 over denominator square root of 34 end fraction space equals negative fraction numerator 2 over denominator square root of 34 end fraction equals fraction numerator 2 over denominator square root of 34 end fraction$

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Three Dimensional Geometry

A, B, C, D are the points (1, – 1, 0), (2, 1, – 1), (– 3, 2, 2) and (0, – 2, – 1) respectively. Find the projection of AB on CD.

Some More Questions From Three Dimensional Geometry Chapter

If α, β, γ are the angles which a line makes with the axes, prove that sin²α + sin² β + sin²γ = 2.

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For Daily Free Study Material Join wiredfaculty

Three Dimensional Geometry

A, B, C, D are the points (1, – 1, 0), (2, 1, – 1), (– 3, 2, 2) and (0, – 2, – 1) respectively. Find the projection of AB on CD.

Some More Questions From Three Dimensional Geometry Chapter

If α, β, γ are the angles which a line makes with the axes, prove that sin2α + sin2 β + sin2γ = 2.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

If α, β, γ are the angles which a line makes with the axes, prove that sin²α + sin² β + sin²γ = 2.