Show that the lines x = ay + b, z = cy + d and x = a' y + b' , z = c' y + d' are perpendicular to each other, if aa' + cc' + 1 = 0.
The equations of the lines are x = ay + b, z = cy + d and or and ∴ direction ratios of two lines are a, 1, c and a', 1, c'. The two lines are perpendicular if (a) (a') + (1) (1) + (c) (c') = 0 [∵ a1 a2 + b1 b2 + c1 c2 = 0] i.e. if aa' + cc' + 1 = 0
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83.If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (– 4, 3, –6) and (2, 9, 2) respectively, then find the angle between the lines AB and CD.
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Long Answer Type
84.Find the equations of the line passing through the point P(–1, 3, –2) and perpendicular to the lines
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85.Find the vector equation of the line passing through the point (1, 2, – 4) and perpendicular to the two lines:
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86.Show that, if the axes are rectangular, then the equations of the line through (α, β, γ) right angles to the lines
are
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87.If the straight lines having direction cosines given by al + bm + cn = 0 and fmn + gnI + hIm = 0 are perpendicular, then show that
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Short Answer Type
88.Find the coordinates of the foot of perpendicular drawn from the point A (1, 2, 1) to the line joining B (1, 4, 6) and (5, 4, 4). Also find the perpendicular distance of A from line BC.
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89.Find the co-ordinates of the foot of the perpendicular drawn from the point A(1, 8, 4) to the line joining the points B (0, –1, 3) and C (2, – 3, – 1).
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Long Answer Type
90.Find the foot of perpendicular from the point (0, 2, 3) on the line
Short Answer Type
Long Answer Type
are prependicular, then find the value of k.
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