Find the condition that the vectors  from Mathematics Vector A

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 Multiple Choice QuestionsShort 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


Here,   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 comma space 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
Let straight a with rightwards arrow on top space and space straight b with rightwards arrow on top be parallel.
therefore space  there exists a non-zero scalar straight lambda such that straight a with rightwards arrow on top space equals space straight lambda space straight b with rightwards arrow on top
therefore space space space straight k space straight i with hat on top space plus space straight l space straight j with hat on top space equals space straight lambda space li with rightwards arrow on top space plus space λk straight j with rightwards arrow on top
therefore space space left parenthesis straight k minus straight lambda space straight l right parenthesis space straight i with hat on top space plus space left parenthesis straight l minus straight lambda space straight k right parenthesis space straight j with hat on top space equals space 0 with rightwards arrow on top
rightwards double arrow space space straight k space minus space straight lambda space straight l space equals space 0 comma space space space space straight l space minus straight lambda space straight k space equals space 0
rightwards double arrow space space space space straight lambda space equals space straight k over straight l comma space straight lambda space equals space straight l over straight k
Equating values of straight lambda comma we get,
                         straight k over straight l space equals space straight l over straight k space space space or space space space straight k squared space equals space straight l squared
which is required condition. 

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 Multiple Choice QuestionsLong 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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 Multiple Choice QuestionsShort 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
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