Find a vector of magnitude 5 units, and parallel to the resultan
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    Vector Algebra

    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.

          straight a with rightwards arrow on top space equals space 2 straight i with hat on top plus 3 straight j with hat on top space minus straight k with hat on top comma space space space space space space space straight b with rightwards arrow on top space equals space straight i with hat on top space minus space 2 straight j with hat on top space plus space straight k with hat on top
    therefore space space space space space straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top space equals space 3 straight i with hat on top space plus space straight j with hat on top therefore space space space space 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 square root of left parenthesis 3 right parenthesis squared plus left parenthesis 1 right parenthesis squared end root space equals space square root of 9 plus 1 end root space equals space square root of 10 therefore space space space straight a space unit space vector space parallel space to space straight a with rightwards arrow on top plus straight b with rightwards arrow on top space is space space equals space fraction numerator straight a with rightwards arrow on top plus straight b with rightwards arrow on top over denominator open vertical bar straight a with rightwards arrow on top space plus straight b with rightwards arrow on top close vertical bar end fraction space equals fraction numerator 1 over denominator square root of 10 end fraction left parenthesis 3 straight i with hat on top space plus space straight j with hat on top right parenthesis space equals space open parentheses fraction numerator 3 over denominator square root of 10 end fraction straight i with hat on top space plus space fraction numerator 1 over denominator square root of 10 end fraction straight j with hat on top close parentheses
    Required vector  = 5 open parentheses fraction numerator straight a with rightwards arrow on top plus straight b with rightwards arrow on top over denominator open vertical bar straight a with rightwards arrow on top plus straight b with rightwards arrow on top close vertical bar end fraction close parentheses space equals space 5 space open parentheses fraction numerator 3 over denominator square root of 10 end fraction straight i with hat on top space plus space fraction numerator 1 over denominator square root of 10 end fraction straight j with hat on top close parentheses

                              equals space fraction numerator 15 over denominator square root of 10 end fraction straight i with hat on top space plus space fraction numerator 5 over denominator square root of 10 end fraction straight j with hat on top space equals space 3 over 2 square root of 10 space straight i with hat on top space plus space 1 half square root of 10 space straight j 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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    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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