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Vector Algebra

Short Answer Type

81. Does there exist scalars u, v, w such that

straight u space stack straight e subscript 1 with rightwards arrow on top space plus space straight v space stack straight e subscript 2 with rightwards arrow on top space plus space straight w space stack straight e subscript 3 with rightwards arrow on top space equals space straight i with hat on top

where

stack straight e subscript 1 with rightwards arrow on top space equals space straight k with hat on top comma space space space stack straight e subscript 2 with rightwards arrow on top space equals space straight j with hat on top space plus space straight k with hat on top comma space space stack straight e subscript 3 with rightwards arrow on top space equals space minus straight j with hat on top space plus space 2 space straight k with hat on top

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82. If the vertices of a triangle are the points with position vectors

straight a subscript 1 straight i with hat on top space plus space straight a subscript 2 straight j with hat on top space plus space straight a subscript 3 straight k with hat on top comma space space space space straight b subscript 1 straight i with hat on top space plus space straight b subscript 2 straight j with hat on top space plus space straight b subscript 3 straight k with hat on top comma space space straight c subscript 1 straight i with hat on top space plus space straight c subscript 2 straight j with hat on top space plus space straight c subscript 3 straight k with hat on top comma space space

what are the vectors determined by its sides? Find the length of these vectors.

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83. Show that the vectors and

straight a with rightwards arrow on top space equals space 3 space straight i with hat on top space minus space 4 space straight j with hat on top space minus space 4 space straight k with hat on top comma space space space space straight b with rightwards arrow on top space equals space 2 straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top comma space space straight c with rightwards arrow on top space equals space straight i with hat on top space minus space 3 space straight j with hat on top space minus space 5 space straight k with hat on top

form the sides of right angled triangle.

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Long Answer Type

84. Show that the points A, B and C with position vectors, respectively form the vertices of a right angled triangle,

stack space straight a with rightwards arrow on top space equals space 3 space straight i with hat on top space minus space 4 space straight j with hat on top space minus space 4 space straight k with hat on top comma space space straight b with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top space and space straight c with rightwards arrow on top space equals space straight i with hat on top space minus space 3 space straight j with hat on top space space minus space 5 space straight k with hat on top comma space

respectively form the vertices of a right angled triangle.

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85. Prove that the points

straight i with hat on top space minus space straight j with hat on top comma space space 4 straight i with hat on top space minus 3 space straight j with hat on top space plus space straight k with hat on top space and space 2 space straight i with hat on top space minus space 4 space straight j with hat on top space plus space 5 space space straight k with hat on top

are the vertices of a right angled triangle.

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86. Show that the points with position vectors

2 space straight i with hat on top space plus space 6 space straight j with hat on top space plus space 3 space straight k with hat on top comma space space straight i with hat on top space plus space 2 space straight j with hat on top space plus space 7 space straight k with hat on top

and

3 straight i with hat on top space plus space 10 space straight j with hat on top space minus space straight k with hat on top

are collinear.

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Short Answer Type

87. Show that the points A (2, 6, 3). B (1, 2, 7) and C (3, 10, – 1) are collinear.

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88. The position vectors of the points A, B and C are $2 space straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top comma space space space space 3 space straight i with hat on top space minus space 2 space straight j with hat on top space plus space straight k with hat on top comma space space space straight i with hat on top space plus space 4 space straight j with hat on top space minus space 3 space straight k with hat on top$ respectively. Show that A, B and C are collinear.

Let $straight a with rightwards arrow on top space equals space 2 straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top comma space space straight b with rightwards arrow on top space equals space 3 space straight i with hat on top space minus space 2 space straight j with hat on top space plus space straight k with hat on top comma space space straight c with rightwards arrow on top space equals space straight i with hat on top space plus space 4 space straight j with hat on top space minus space 3 space straight k with hat on top$ be position vectors of A, B and C respectively.
$AB with rightwards arrow on top space equals space straight P. straight V. space of space straight B space minus space straight P. straight V. space of space straight A space equals space straight b with rightwards arrow on top space minus space straight a with rightwards arrow on top space equals space open parentheses 3 straight i with hat on top space minus space 2 straight j with hat on top space plus space straight k with hat on top close parentheses space minus space left parenthesis 2 space straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top right parenthesis$
$equals space straight i with hat on top space minus space 3 space straight j with hat on top space plus space 2 space straight k with hat on top$

$AC with rightwards arrow on top space equals space straight P. straight V. space of space straight C space minus space straight P. straight V. space of space straight A space equals space straight c with rightwards arrow on top space minus space straight a with rightwards arrow on top space equals space space open parentheses straight i with hat on top space plus space 4 space straight j with hat on top space minus space 3 space straight k with hat on top close parentheses space minus space left parenthesis 2 straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top right parenthesis space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals negative space straight i with hat on top space plus space 3 space straight j with hat on top space minus space 2 space straight k with hat on top therefore space space space space AC with rightwards arrow on top space equals space minus AB with rightwards arrow on top therefore space space space space AC with rightwards arrow on top space and space AB with rightwards arrow on top space are space parallel space vectors But space straight A space is space their space common space point. therefore space space space points space straight A comma space straight B comma space straight C space are space collinear.$