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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.

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89. The position vectors of the points A, B, C and D are

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

respectively. Show that AB and CD are parallel.

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90. Find the direction cosines of the vector $straight i with hat on top space plus space 2 space straight j with hat on top space plus space 3 space straight k with hat on top.$

Let $straight a with rightwards arrow on top space equals space straight i with hat on top space plus space 2 space straight j with hat on top space plus space 3 space straight k with hat on top$

$therefore space space open vertical bar straight a with rightwards arrow on top close vertical bar space equals space square root of left parenthesis 1 right parenthesis squared plus left parenthesis 2 right parenthesis squared plus left parenthesis 3 right parenthesis squared end root space equals space square root of 1 plus 4 plus 9 end root space equals space square root of 14 therefore space space space space space straight a with hat on top space equals space fraction numerator open vertical bar straight a with rightwards arrow on top close vertical bar over denominator open vertical bar straight a with rightwards arrow on top close vertical bar end fraction space equals space fraction numerator 1 over denominator square root of 14 end fraction left parenthesis straight i with hat on top space plus space 2 space straight j with hat on top space plus space 3 space straight k with hat on top right parenthesis space space space space space space space space space space space space equals space fraction numerator 1 over denominator square root of 14 end fraction straight i with hat on top space plus space fraction numerator 2 over denominator square root of 14 end fraction straight j with hat on top space plus space fraction numerator 3 over denominator square root of 14 end fraction straight k with hat on top therefore space space space direction space cosines space of space straight a with rightwards arrow on top space are space fraction numerator 1 over denominator square root of 14 end fraction comma space fraction numerator 2 over denominator square root of 14 end fraction comma space fraction numerator 3 over denominator square root of 14 end fraction.$

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