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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 $square root of 3$

Let $straight a with rightwards arrow on top space equals space straight x subscript 1 straight i with hat on top space plus space straight y subscript 1 straight j with hat on top space plus space straight z subscript 1 straight k with hat on top comma space space straight b with rightwards arrow on top space equals space straight a with rightwards arrow on top space equals straight x subscript 2 straight i with hat on top space space plus space straight y subscript 2 space straight j with hat on top space plus space straight z subscript 2 straight k with hat on top$ be two units vectors such that
$straight a with rightwards arrow on top plus straight b with rightwards arrow on top$ is also a unit vector.
$therefore space space space space open vertical bar straight a with rightwards arrow on top close vertical bar space equals space 1 comma space space space open vertical bar straight b with rightwards arrow on top close vertical bar space equals space 1 comma space space space open vertical bar straight a with rightwards arrow on top plus straight b with rightwards arrow on top close vertical bar space equals space 1$
$open vertical bar straight a with rightwards arrow on top close vertical bar space equals space 1 space space space space rightwards double arrow space space space space square root of straight x subscript 1 squared plus straight y subscript 1 squared plus straight z subscript 1 squared end root space equals space 1 space space space space space rightwards double arrow space space space space straight x subscript 1 squared plus space straight y subscript 1 squared plus straight z subscript 1 squared space equals space 1 space space space space space space... left parenthesis 1 right parenthesis open vertical bar straight b with rightwards arrow on top close vertical bar space equals space 1 space rightwards double arrow space space space square root of straight x subscript 2 squared plus straight y subscript 2 squared plus straight z subscript 2 squared end root space equals space 1 space space space space space space space space space space rightwards double arrow space space straight x subscript 2 squared plus straight y subscript 2 squared plus straight z subscript 2 squared space equals space 1 space space space space space space space space space space... left parenthesis 2 right parenthesis open vertical bar straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top close vertical bar space equals space 1 space space rightwards double arrow space space space space square root of left parenthesis straight x subscript 1 plus straight x subscript 2 right parenthesis squared space plus space left parenthesis straight y subscript 1 plus straight y subscript 2 right parenthesis squared plus left parenthesis straight z subscript 1 plus straight z subscript 2 right parenthesis squared end root space equals space 1$
$rightwards double arrow space space space space space left parenthesis straight x subscript 1 plus straight x subscript 2 right parenthesis squared plus space left parenthesis straight y subscript 1 plus straight y subscript 2 right parenthesis squared space plus space left parenthesis straight z subscript 1 plus straight z subscript 2 right parenthesis squared space equals space 1 rightwards double arrow space space space space left parenthesis straight x subscript 1 squared plus straight y subscript 1 squared plus straight z subscript 1 squared right parenthesis space plus space left parenthesis straight x subscript 2 squared plus straight y subscript 2 squared plus straight z subscript 2 squared right parenthesis space plus space 2 left parenthesis straight x subscript 1 straight x subscript 2 plus straight y subscript 1 straight y subscript 2 plus straight y subscript 1 plus straight z subscript 1 straight z subscript 2 right parenthesis space equals 1 rightwards double arrow space space space 1 plus 1 plus 2 space left parenthesis straight x subscript 1 straight x subscript 2 space plus space straight y subscript 1 straight y subscript 2 plus straight z subscript 1 straight z subscript 2 right parenthesis space equals space minus 1 space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 3 right parenthesis Now comma space open vertical bar straight a with rightwards arrow on top minus space straight b with rightwards arrow on top close vertical bar space equals space square root of left parenthesis straight x subscript 1 minus straight x subscript 2 right parenthesis squared plus left parenthesis straight y subscript 1 minus straight y subscript 2 right parenthesis squared plus left parenthesis straight z subscript 1 minus straight z subscript 2 right parenthesis squared end root space space space space equals space square root of left parenthesis straight x subscript 1 squared plus straight y subscript 1 squared plus straight y subscript 1 squared right parenthesis space plus space left parenthesis straight x subscript 2 squared plus straight y subscript 2 squared plus straight z subscript 2 squared right parenthesis space minus space 2 left parenthesis straight x subscript 1 straight x subscript 2 plus straight y subscript 1 straight y subscript 2 plus straight z subscript 1 straight z subscript space 2 end subscript right parenthesis end root space space space space equals space square root of 1 plus 1 plus 1 end root space equals space square root of 3 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left square bracket because space of space left parenthesis 1 right parenthesis comma space left parenthesis 2 right parenthesis comma space left parenthesis 3 right parenthesis right square bracket space space space space space space space space space space space space space space space space space space space space space space space space space space space$