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Three Dimensional Geometry

Short Answer Type

101.

Find the area of the triangle whose vertices are (1, 2, 4), (-2, 1, 2), (2, 4, -3).

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Long Answer Type

102. A line makes angle α, β, γ and δ with the diagonals of a cube, prove that

cos squared straight alpha space plus space cos squared straight beta space plus space cos squared straight gamma space space plus cos squared straight delta space equals space 4 over 3.

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103. If the edges of a rectangular parallelepiped are a, b, c, show that the angles between four diagonals are given by cos^–1

open parentheses fraction numerator straight a squared plus-or-minus straight b squared plus-or-minus straight c squared over denominator straight a squared plus straight b squared plus straight c squared end fraction close parentheses.

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104. Find the angle between two diagonals of a cube.

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105. Show that the line joining the middle points of two sides of a triangle is parallel to the third side and half of it in length.

Let A (x₁, y₁, z₁). B (x₂, y₂, z₂), C (x₃, y₃, z₃) be the vertices of Δ ABC and D, E, F be mid-points of BC, CA and AB respectively.

therefore space space space straight E space is space open parentheses fraction numerator straight x subscript 3 plus straight x subscript 1 over denominator 2 end fraction comma space fraction numerator straight y subscript 3 plus straight y subscript 1 over denominator 2 end fraction comma space fraction numerator straight z subscript 3 plus straight z subscript 1 over denominator 2 end fraction close parentheses

and

straight F space is space open parentheses fraction numerator straight x subscript 1 plus straight x subscript 2 over denominator 2 end fraction comma space fraction numerator straight y subscript 1 plus straight y subscript 2 over denominator 2 end fraction comma space fraction numerator straight z subscript 1 plus straight z subscript 2 over denominator 2 end fraction close parentheses

Direction-ratios of BC are
x₃– x₂, y₃– y₂, z₃ – z₂.
Directions-ratios of FE are
$fraction numerator straight x subscript 3 plus straight x subscript 1 over denominator 2 end fraction minus fraction numerator straight x subscript 1 plus straight x subscript 2 over denominator 2 end fraction comma space space fraction numerator straight y subscript 3 plus straight y subscript 1 over denominator 2 end fraction minus fraction numerator straight y subscript 1 plus straight y subscript 2 over denominator 2 end fraction comma space space fraction numerator straight z subscript 3 plus straight z subscript 1 over denominator 2 end fraction minus fraction numerator straight z subscript 1 plus straight z subscript 2 over denominator 2 end fraction$

or $fraction numerator straight x subscript 3 minus straight x subscript 2 over denominator 2 end fraction comma space space fraction numerator straight y subscript 3 minus straight y subscript 2 over denominator 2 end fraction comma space space fraction numerator straight z subscript 3 minus straight z subscript 2 over denominator 2 end fraction$

or $straight x subscript 3 minus straight x subscript 2 comma space space straight y subscript 3 minus straight y subscript 2 comma space space straight z subscript 3 minus straight z subscript 2$

which are the same as that of BC
∴ FE || BC.
Also,
$FE space equals space square root of open parentheses fraction numerator straight x subscript 3 plus straight x subscript 1 over denominator 2 end fraction minus space fraction numerator straight x subscript 1 plus straight x subscript 2 over denominator 2 end fraction close parentheses squared plus open parentheses fraction numerator straight y subscript 3 plus straight y subscript 1 over denominator 2 end fraction minus fraction numerator straight y subscript 1 plus straight y subscript 2 over denominator 2 end fraction close parentheses squared space plus space open parentheses fraction numerator straight z subscript 3 plus straight z subscript 1 over denominator 2 end fraction minus fraction numerator straight z subscript 1 plus straight z subscript 2 over denominator 2 end fraction close parentheses squared end root space space space space space equals space 1 half square root of left parenthesis straight x subscript 3 minus straight x subscript 2 right parenthesis squared plus left parenthesis straight y subscript 3 minus straight y subscript 2 right parenthesis squared plus left parenthesis straight z subscript 3 minus straight z subscript 2 right parenthesis squared end root space equals space 1 half BC$
Hence the result.

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106. A variable line in two adjacent positions has direction cosines < l, m, n > and < l + δl, m + δm, n + δn >. Show that the small angle δθ between two positions is given by
(δθ )² = (δl)² + (δm)² + (δn)²

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107.

Find the angle between the two lines whose direction cosines are given by the equations:
l + m + n = 0, l² + m² – n² = 0

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108. Find the angle between the two lines whose direction cosines are given by the equations:
2 l – m + 2 n = 0 and m n + n l + l m = 0

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109. Find the angle between the two lines whose direction cosines are given by the equations:
l + m + n = 0 and 2 l + 2 m – m n = 0

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110. Show that the straight lines whose direction cosines are given by the equations uI + vm + wn = 0, a I² + b m² + cn² = 0 are
(i) perpendicular if u² (b + c) + v² (c + a) + w² (a + b) = 0
(ii) parallel if

straight u squared over straight a plus straight v squared over straight b plus straight w squared over straight c equals 0.

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