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Vector Algebra

Question
CBSEENMA12034125

Let straight a with rightwards arrow on top space equals straight i with hat on top plus 4 straight j with hat on top plus 2 straight k with hat on top comma space space straight b with rightwards arrow on top space equals space 3 straight i with hat on top space minus space 2 straight j with hat on top space plus space 7 straight k with hat on top space and space straight c with rightwards arrow on top space equals space 2 straight i with hat on top minus straight j with hat on top minus 4 straight k with hat on top. Find a vector straight d with rightwards arrow on top which is perpendicular to both straight a with rightwards arrow on top space and space straight b with rightwards arrow on top space and space straight c with rightwards arrow on top. space straight d with rightwards arrow on top space equals space 15.

Solution

Here straight a with rightwards arrow on top space equals space straight i with hat on top plus 4 straight j with hat on top plus 2 straight k with hat on top comma space space straight b with rightwards arrow on top space equals 3 straight i with hat on top minus 2 straight j with hat on top plus 7 straight k with hat on top comma space straight c with rightwards arrow on top space equals space 2 straight i with hat on top minus straight j with hat on top minus 4 straight k with hat on top
Let straight d with rightwards arrow on top space equals space straight d subscript 1 straight i with hat on top plus straight d subscript 2 straight j with hat on top plus straight d subscript 3 straight k with hat on top
Since straight d with rightwards arrow on top is perpendicular to straight a with rightwards arrow on top space and space straight b with rightwards arrow on top
therefore space space straight d with rightwards arrow on top. space straight a with rightwards arrow on top space equals 0 space space space and space straight d with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 0
therefore space space space space space straight d subscript 1 plus 4 straight d subscript 2 space plus space 2 straight d subscript 3 space equals space 0                                               ...(1)
and    3 space straight d subscript 1 space minus space 2 space straight d subscript 2 space plus space 7 space straight d subscript 3 space equals space 0                                      ...(2)
Again straight c with rightwards arrow on top. straight d with rightwards arrow on top space equals space 15
therefore space space space 2 space straight d subscript 1 space minus space straight d subscript 2 space plus space 4 space straight d subscript 3 space equals space 15                                         ...(3)
Multiplying (1) by 7 and (2) by -2, we get,
                  7 space straight d subscript 1 space plus space 28 space straight d subscript 2 space plus space 14 space straight d subscript 3 space equals space 0
minus 6 space straight d subscript 1 space plus space 4 space straight d subscript 2 space minus space 14 space straight d subscript 3 space equals space 0
Adding,                 straight d subscript 1 space plus space 32 space straight d subscript 2 space equals space 0                               ...(4)
Multiplying (1) by 2  and (3) by -1, we get,
               2 space straight d subscript 1 plus space 8 space straight d subscript 2 space plus space 4 space straight d subscript 3 space equals space 0
minus 2 space straight d subscript 1 plus space straight d subscript 2 space minus space 4 space straight d subscript 3 space equals space 0
Adding,          9 space straight d subscript 2 space equals space minus 15 comma space space space space space therefore space space straight d subscript 2 space equals space minus 15 over 9 space equals space minus 5 over 3
From (4),   straight d subscript 1 minus 160 over 3 space equals space 0 comma            therefore space space space straight d subscript 1 space equals space 160 over 3
From (1),    160 over 3 minus 20 over 3 plus 2 space straight d subscript 3 space equals space 0 comma space space space space space space therefore space space space 2 space straight d subscript 3 space equals space 20 over 3 minus 160 over 3
therefore space space 2 space straight d subscript 3 space equals space fraction numerator 20 minus 160 over denominator 3 end fraction space equals space minus 140 over 3 space space space space space space rightwards double arrow space space space space straight d subscript 3 space equals space minus 70 over 3
therefore space space required space vector space straight d with rightwards arrow on top space equals space 160 over 3 straight i with hat on top minus 5 over 3 straight j with hat on top space minus 70 over 3 straight k with hat on top
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 space space space equals space 5 over 3 left parenthesis 32 space straight i with hat on top space minus space straight j with hat on top space minus space 14 space straight k with hat on top right parenthesis