If the vertices of a triangle are the points with position vecto

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 Multiple Choice QuestionsShort 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. 


Let A, B, C be vertices of given triangle with position vectors straight a with rightwards arrow on top space equals straight a subscript 1 space 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,
straight b with rightwards arrow on top space equals space straight b subscript 1 space straight i with hat on top space plus space straight b subscript 2 space straight j with hat on top space plus space straight b subscript 3 straight k with hat on top comma space space straight c with rightwards arrow on top space equals 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 space respectively. space
 Now comma space space space space 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 space space space space space equals space left parenthesis straight b subscript 1 space minus space straight a subscript 1 right parenthesis straight i with hat on top space plus space left parenthesis straight b subscript 2 minus straight a subscript 2 right parenthesis space straight j with hat on top space plus space left parenthesis straight b subscript 3 minus straight a subscript 3 right parenthesis space straight k with rightwards arrow on top
BC 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 B space equals space straight c with rightwards arrow on top space minus space straight b with rightwards arrow on top
space space space space space space space equals space left parenthesis straight c subscript 1 minus straight b subscript 1 right parenthesis space space straight i with hat on top space plus space left parenthesis straight c subscript 2 minus straight b subscript 2 right parenthesis space straight j with hat on top space plus space left parenthesis straight c subscript 3 minus straight b subscript 3 right parenthesis space straight k with hat on top
CA with rightwards arrow on top space equals space straight P. straight V. space of space straight A space minus space straight P. straight V. space of space straight C space equals space straight a with rightwards arrow on top space minus space straight c with rightwards arrow on top
space space space space space space equals space left parenthesis straight a subscript 1 space minus space straight c subscript 1 right parenthesis space straight i with hat on top space plus space left parenthesis straight a subscript 2 minus straight c subscript 2 right parenthesis space straight j with hat on top space plus space left parenthesis straight a subscript 3 minus straight c subscript 3 right parenthesis space straight k with hat on top

therefore space space space AB space equals space square root of left parenthesis straight b subscript 1 minus straight a subscript 1 right parenthesis squared plus left parenthesis straight b subscript 2 minus straight a subscript 2 right parenthesis squared plus left parenthesis straight b subscript 3 minus straight a subscript 3 right parenthesis squared end root
space space space space space space BC space equals space square root of left parenthesis straight c subscript 1 minus straight b subscript 1 right parenthesis squared space plus space left parenthesis straight c subscript 2 minus straight b subscript 2 right parenthesis squared plus left parenthesis straight c subscript 3 minus straight b subscript 3 right parenthesis squared end root
space space space space space CA space equals space square root of left parenthesis straight a subscript 1 minus straight c subscript 1 right parenthesis squared plus left parenthesis straight a subscript 2 minus straight c subscript 2 right parenthesis squared plus left parenthesis straight a subscript 3 minus straight c subscript 3 right parenthesis squared end root

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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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 Multiple Choice QuestionsLong 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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 Multiple Choice QuestionsShort 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.
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