Prove that the points are the vertices of a right angled tria

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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.

Let A, B, C, be the points whose position vectors are

straight a with rightwards arrow on top space equals space straight i with hat on top space minus space straight j with hat on top comma space space space straight b with rightwards arrow on top space equals space 4 space straight i with hat on top space minus space 3 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 2 space straight i with hat on top space space minus space 4 space straight j with hat on top space plus space 5 space straight k with hat on top 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 space space equals space left parenthesis 4 space straight i with hat on top space minus space 3 space straight j with hat on top space plus space space straight k with hat on top right parenthesis space minus space left parenthesis straight i with hat on top space minus space space straight j with hat on top right parenthesis 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 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 space equals space left parenthesis 2 straight i with hat on top space minus space 4 space straight j with hat on top space plus space 5 space straight k with hat on top right parenthesis space minus space left parenthesis 4 space straight i with hat on top space minus space 3 space straight j with hat on top space plus space straight k with hat on top right parenthesis space equals space minus space 2 straight i with hat on top space minus space straight j with hat on top space plus space 4 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 space space space space space equals left parenthesis straight i with hat on top space minus space straight j with hat on top right parenthesis space minus space left parenthesis 2 straight i with hat on top space minus space 4 space straight j with hat on top space plus space space 5 space straight k with hat on top right parenthesis space equals space minus straight i with hat on top space plus space 3 space straight j with hat on top space minus space 5 space straight k with hat on top therefore space space space AB with rightwards arrow on top space plus space BC with rightwards arrow on top space plus space CA with rightwards arrow on top space equals space 0 with rightwards arrow on top therefore space space space ABC space is space straight a space triangle.

AB space equals space square root of 9 plus 4 plus 1 end root space equals space square root of 14 BC space equals space square root of 4 plus 1 plus 16 end root space equals space square root of 21 CA space equals space square root of 1 plus 9 plus 25 end root space equals square root of 35

Now,

AB squared plus BC squared space equals space CA squared

therefore space space space increment ABC space is space right space angled space at space straight B.

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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.

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