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Three Dimensional Geometry

Short Answer Type

91. Find the length of perpendicular from (2, 1, 3) on the line

fraction numerator straight x minus 4 over denominator 5 end fraction space equals fraction numerator straight y minus 2 over denominator 4 end fraction space equals space fraction numerator straight z minus 3 over denominator 3 end fraction.

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92.

Find the length and foot of the perpendicular drawn from the point (3, 4, 5) on the line
$fraction numerator straight x minus 2 over denominator 2 end fraction space equals space fraction numerator straight y minus 3 over denominator 5 end fraction space equals space fraction numerator straight z minus 1 over denominator 3 end fraction$

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93.

Find the perpendicular distance of the point (1, 0, 0) form the line $fraction numerator straight x minus 1 over denominator 2 end fraction space equals space fraction numerator straight y plus 1 over denominator negative 3 end fraction space equals space fraction numerator straight z plus 10 over denominator 8 end fraction$

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94.

Find the length of the perpendicular drawn from the point (1, ,2 3) on the line
$fraction numerator straight x minus 6 over denominator 3 end fraction space equals space fraction numerator straight y minus 7 over denominator 2 end fraction space equals space fraction numerator straight z minus 7 over denominator negative 2 end fraction$

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95. Find the length and the foot of the perpendicular drawn from the point (2, – 1, 5) to the line

fraction numerator straight x minus 11 over denominator 10 end fraction space equals space fraction numerator straight y plus 2 over denominator negative 4 end fraction space equals space fraction numerator straight z plus 8 over denominator negative 11 end fraction.

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Long Answer Type

96. Find the equation of the perpendicular drawn from the point (2, 4, – 1) to the line

fraction numerator straight x plus 5 over denominator 1 end fraction space equals space fraction numerator straight y plus 3 over denominator 4 end fraction space equals space fraction numerator straight z minus 6 over denominator negative 9 end fraction.

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97. Find the length of perpendicular from (3, 2, 1) on the line

fraction numerator straight x minus 4 over denominator 5 end fraction space equals space fraction numerator straight y minus 2 over denominator 3 end fraction space equals space fraction numerator straight z minus 3 over denominator 4 end fraction

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Short Answer Type

98. Find the coordinates of the foot of perpendicular drawn from the point A (1, 2, 1) to the line

fraction numerator straight x minus 1 over denominator 4 end fraction space equals space fraction numerator straight y minus 4 over denominator 0 end fraction space equals space fraction numerator straight z minus 6 over denominator negative 2 end fraction

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Long Answer Type

99. Find the image of the point (1, 6, 3) in the line

straight x over 1 space equals space fraction numerator straight y minus 1 over denominator 2 end fraction space equals space fraction numerator straight z minus 2 over denominator 3 end fraction.

416 Views

Short Answer Type

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

Let A (1, – 1, – 3), B (4, – 3, 1), C (3, – 1, 2) be vertices of Δ ABC

AB space equals space square root of left parenthesis 4 minus 1 right parenthesis squared plus left parenthesis negative 3 plus 1 right parenthesis squared plus left parenthesis 1 plus 3 right parenthesis squared end root space equals space square root of 9 plus 4 plus 16 end root space equals space square root of 29 AC space equals space square root of left parenthesis 3 minus 1 right parenthesis squared plus left parenthesis negative 1 plus 1 right parenthesis squared plus left parenthesis 2 plus 3 right parenthesis squared end root space equals space square root of 4 plus 0 plus 25 end root space equals space square root of 29

Direction-ratios of AB are 4 – 1, – 3 + 1, 1 + 3 i.e.. 3, – 2, 4 respectively.
Direction-ratios of AC are 3 – 1, –1 + 1, 2 + 3 i.e., 2, 0, 5 respectively.
$therefore space space space cos space straight A space equals space fraction numerator left parenthesis 3 right parenthesis thin space left parenthesis 2 right parenthesis space plus space left parenthesis negative 2 right parenthesis space left parenthesis 0 right parenthesis space plus space left parenthesis 4 right parenthesis thin space left parenthesis 5 right parenthesis over denominator square root of 9 plus 4 plus 16 end root space square root of 4 plus 0 plus 25 end root end fraction space equals space fraction numerator 6 plus 0 plus 20 over denominator square root of 29 space square root of 29 end fraction space equals space 26 over 29 therefore space space sin space straight A space equals space square root of 1 minus space cos squared straight A end root space equals space square root of 1 minus open parentheses 26 over 29 close parentheses squared end root space equals space square root of 1 minus 676 over 841 end root space equals space square root of 165 over 841 end root space equals space fraction numerator square root of 165 over denominator 29 end fraction$
Area of $increment ABC space equals space 1 half AB. space AC space sin space straight A space equals space 1 half square root of 29. space square root of 29. space fraction numerator square root of 165 over denominator 29 end fraction space equals space 1 half square root of 165 space sq. space units. space$

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