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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.

Take O, a corner of cube OBLCMANP, as origin and OA, OB, OC, the three edges through it as the axes.

Let OA = OB = OC = a, then the co-ordinates of O, A, B, C are (0, 0, 0), (a, 0, 0), (0, a, 0), (0, 0, a) respectively ; those of P, L, M, N are (a, a, a), (0, a, a), (a, 0, a), (a, a, 0) respectively.
The four diagonals are OP, AL, BM, CN. Direction cosines of OP are proportional to a – 0, a – 0, a – 0, i.e., a, a, a, i.e., 1, 1, 1.
Direction-cosines of AL are proportional to 0 – a, a – 0, a – 0    i.e., –a, a, a, i.e., – 1, 1, 1.
Direction-cosines of BM are proportional to a – 0, 0 – a, a – 0. i.e.. a – a, a    i.e., 1, – 1, 1.
Direction-cosines of CN are proportional to a – 0, a – 0, 0 – a    i.e., a, a, – a    i.e., 1, 1, – 1.
Let θ be the angle between AL and BM
$therefore space space space space space space cos space straight theta space equals space open vertical bar fraction numerator left parenthesis negative 1 right parenthesis thin space left parenthesis 1 right parenthesis space plus space left parenthesis 1 right parenthesis thin space left parenthesis negative 1 right parenthesis space plus space left parenthesis 1 right parenthesis thin space left parenthesis 1 right parenthesis over denominator square root of 1 plus 1 plus 1 end root space square root of 1 plus 1 plus 1 end root end fraction close vertical bar space equals space open vertical bar fraction numerator negative 1 minus 1 plus 1 over denominator square root of 3. square root of 3 end fraction close vertical bar space equals space 1 third therefore space space space space straight theta space equals space cos to the power of negative 1 end exponent 1 third$
Similarly the angle between other two diagonals is also cos $1 third$ .

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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.

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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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