Does there exist scalars u, v, w such that        where  �
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    Vector Algebra

    Multiple Choice QuestionsShort Answer Type

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    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 ?

    Here,   stack straight e subscript 1 with rightwards arrow on top space equals straight k with hat on top comma 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 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
    Now comma space space 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
    rightwards double arrow space space space straight u left parenthesis straight k with hat on top right parenthesis space plus space straight v left parenthesis straight j with hat on top space plus space straight k with hat on top right parenthesis space plus space straight w space open parentheses negative straight j with hat on top space plus space 2 space straight k with hat on top close parentheses space equals straight i with hat on top rightwards double arrow space space 0 straight i with hat on top space plus space left parenthesis straight v minus straight w right parenthesis space straight j with hat on top space plus space left parenthesis straight u plus straight v space plus space 2 straight w right parenthesis space straight k with hat on top space equals space 1 space straight i with hat on top space plus space 0 space straight j with hat on top space plus space 0 space straight k with hat on top rightwards double arrow space space 0 space equals space 1 comma space space space which space is space not space possible therefore space space space there space exist space on space scalars space straight u comma space straight v comma space straight w space such space that space 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

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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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    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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