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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.

Let $straight a with rightwards arrow on top space equals space 2 straight i with hat on top space plus space 6 space straight j with hat on top space plus space space 3 space straight k with hat on top comma space space space straight b with rightwards arrow on top space equals space straight i with hat on top space plus space space 2 straight j with hat on top space plus space 7 space straight k with hat on top comma space space space straight c with rightwards arrow on top space equals space 3 straight i with hat on top space plus 10 straight j with hat on top space minus space straight k with hat on top$ be position vectors of A, B, C respectively.
$AB with rightwards arrow on top space equals space straight P. straight V. space of space straight B space minus space straight P. straight V. space of space straight A space equals space straight b with rightwards arrow on top space minus space straight a with rightwards arrow on top$
$equals left parenthesis straight i with hat on top space plus 2 space straight j with hat on top space plus space space 7 space straight k with hat on top right parenthesis space minus space left parenthesis 2 space straight i with hat on top space plus space 6 space straight j with hat on top space space plus space 3 space straight k with hat on top right parenthesis space equals space minus space straight i with hat on top space minus space 4 space straight j with hat on top space plus space 4 space straight k with hat on top$
$BC with rightwards arrow on top space equals space straight P. straight V. space of space straight C space minus space straight P. straight V. space of space straight B space equals space straight c with rightwards arrow on top space minus space straight b with rightwards arrow on top space space space space space space equals space left parenthesis 3 space straight i with hat on top space plus space 10 space straight j with hat on top space minus space straight k with hat on top right parenthesis space space minus left parenthesis straight i with hat on top space plus space 2 space straight j with hat on top space space plus space 7 space straight k with hat on top right parenthesis space equals space 2 space straight i with hat on top space plus space space 8 space straight j with hat on top space minus space 8 space straight k with hat on top$
$therefore space space space space space BC with rightwards arrow on top space equals space minus space 2 space AB with rightwards arrow on top$
$therefore space space space AB with rightwards arrow on top space and space BC with rightwards arrow on top space are space parallel space vectors. But space straight A space is space their space common space point. space therefore space space space space space points space straight A comma space straight B comma space straight C space are space collinear.$