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Vector Algebra

Short Answer Type

71. For given vectors,

straight a 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 2 space straight k with hat on top space and space straight b with rightwards arrow on top space equals space minus 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

find the unit vector in the direction of the vector

straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top

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72.

If $straight a with rightwards arrow on top space equals straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top$ , $straight b with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space straight j with hat on top space plus space 3 space straight k with hat on top$ and $straight c with rightwards arrow on top space equals space straight i with hat on top space minus space 2 space straight j with hat on top space plus space straight k with hat on top$ , find a vector parallel to the vector $2 space straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top space plus space 3 space straight c with rightwards arrow on top$ .

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73. Find a vector in the direction of vector

5 space straight i with hat on top space minus space straight j with hat on top space plus space 2 space straight k with hat on top

which has magnitude 8 units.

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74. Find a vector of magnitude 5 units, and parallel to the resultant of the vectors

straight a with rightwards arrow on top space equals space 2 straight i with hat on top space plus space 3 straight j with hat on top space minus space straight k with hat on top space and space straight b with rightwards arrow on top space equals 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.

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75. Show that the vectors

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

and

negative 4 space straight i with hat on top space plus space 6 space straight j with hat on top space minus space 8 space straight k with hat on top

are collinear.

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76. Find the condition that the vectors

straight a with rightwards arrow on top space equals space straight k space straight i with hat on top space plus space straight l space straight j with hat on top space and space straight b with rightwards arrow on top space equals space straight l space straight i with hat on top space plus space straight k space straight j with hat on top comma space space left parenthesis straight k comma space straight l space not equal to 0 right parenthesis space are space parallel

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Long Answer Type

77. Prove that if

straight u with rightwards arrow on top space equals space straight u subscript 1 space straight i with hat on top space plus space straight u subscript 2 space straight j with hat on top space and space straight v with rightwards arrow on top space equals space straight v subscript 1 space straight i with hat on top space plus space straight v subscript 2 space straight j with hat on top

are non-zero vectors, then they are parallel if and only if u₂v₂ – u₂v₁ = 0.

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78. Find the condition that the vectors

straight a with rightwards arrow on top space equals space straight k space straight i with hat on top space plus space 3 space straight j with hat on top space and space straight b with rightwards arrow on top space equals space 4 space straight i with hat on top space plus space straight k space straight j with hat on top comma space space left parenthesis straight k space not equal to space 0 right parenthesis

are parallel.

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79. If the sum of two unit vectors is a unit vector, show that the magnitude of their difference is

square root of 3

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Short Answer Type

80. Show that the three points having position vectors
$straight a with rightwards arrow on top space minus space 2 straight b with rightwards arrow on top space space plus space space 3 straight c with rightwards arrow on top comma space space minus 2 space straight a with rightwards arrow on top space plus space 3 space straight b with rightwards arrow on top space plus space 2 space straight c with rightwards arrow on top comma space minus 8 space straight a with rightwards arrow on top space plus space 13 space straight b with rightwards arrow on top$ are collinear whatever be $straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top$

Let $straight a with rightwards arrow on top space minus space 2 straight b with rightwards arrow on top space plus space 3 space straight c with rightwards arrow on top comma space space space minus 2 space straight a with rightwards arrow on top space plus space 3 space straight b with rightwards arrow on top space plus space 2 space straight c with rightwards arrow on top comma space space minus 8 space straight a with rightwards arrow on top space plus space 13 space straight b with rightwards arrow on top$ be the position vectors of A, B, C respectively...
$AB with rightwards arrow on top space equals space straight P. straight V. space of space straight B space minus space straight P. straight V. space of space straight A space equals space minus 2 straight a with rightwards arrow on top space plus space 3 straight b with rightwards arrow on top plus space 2 space straight c with rightwards arrow on top space minus space straight a with rightwards arrow on top space plus space 2 space straight b with rightwards arrow on top space minus space 3 straight c with rightwards arrow on top space equals space minus 3 space straight a with rightwards arrow on top space plus space 5 space straight b with rightwards arrow on top space minus space straight c with rightwards arrow on top$
$AC with rightwards arrow on top space equals space straight P. straight V. space of space straight C space minus space straight P. straight V. space of space straight A space equals space minus 8 straight a with rightwards arrow on top space plus space 13 space straight b with rightwards arrow on top space minus space straight a with rightwards arrow on top space plus space 2 space straight b with rightwards arrow on top space minus space 3 space straight c with rightwards arrow on top space equals space minus 9 straight a with rightwards arrow on top space plus space 15 space straight b with rightwards arrow on top minus 3 space straight c with rightwards arrow on top$
$therefore space space space space space AC with rightwards arrow on top space equals space 3 space AB with rightwards arrow on top space space space space space space space rightwards double arrow space space space space AC with rightwards arrow on top space and space AB with rightwards arrow on top space are space parallel space vectors$
But A is their common point
∴ point A, B, C are collinear.

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Long Answer Type

Short Answer Type

80. Show that the three points having position vectors are collinear whatever be