-->

Vector Algebra

Question
CBSEENMA12034124

Let straight a with rightwards arrow on top space equals space 4 straight i with hat on top space plus space 5 straight j with hat on top space minus space straight k with hat on top comma space straight b with rightwards arrow on top space equals space straight i with hat on top minus 4 straight j with hat on top plus space 5 straight k with hat on top space and space straight c with rightwards arrow on top space equals 3 straight i with hat on top plus straight j with hat on top minus straight k with hat on top. space 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 d with rightwards arrow on top. space straight c with rightwards arrow on top space equals space 21.

Solution
straight a with rightwards arrow on top space equals space 4 straight i with hat on top space plus space 5 straight j with hat on top space minus space straight k with hat on top comma space space straight b with rightwards arrow on top space equals space straight i with hat on top space minus space 4 straight j with hat on top space plus space 5 straight k with hat on top comma space straight c with rightwards arrow on top space equals space 3 straight i with hat on top space plus space straight j with hat on top space minus space 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. space straight a with rightwards arrow on top space equals space 0 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 4 space straight d subscript 1 space plus space 5 space straight d subscript 2 space minus space straight d subscript 3 space equals space 0 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 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 parenthesis 1 right parenthesis
and   straight d subscript 1 minus 4 straight d subscript 2 plus 5 straight d subscript 3 space equals 0                                                                     ...(2)
Again,                  straight d with rightwards arrow on top. space straight i with hat on top space equals space 21
rightwards double arrow space space space 3 space straight d subscript 1 space plus space straight d subscript 2 space minus straight d subscript 3 space equals space 21                                                                 ...(3)
Subtracting (3) from (1), we get, 
                          straight d subscript 1 plus 4 space straight d subscript 2 space equals space minus 21                                                       ...(4)               
Multiply (2) by 1 and (3) by 5, we get.
    straight d subscript 1 minus 4 space straight d subscript 2 plus 5 space straight d subscript 3 space equals 0
   15 space straight d subscript 1 space plus space 5 space straight d subscript 2 space minus space 5 space straight d subscript 3 space equals space 105
Adding,      16 space straight d subscript 1 space plus space straight d subscript 2 space equals space 105                                                           ...(5)
therefore space space space space 64 space straight d subscript 1 space plus space 4 space straight d subscript 2 space equals space 420                                                                      
Subtracting it from (4), we get,
                    negative 63 space straight d subscript 1 space equals space minus 441 space space space space space space space space space space space rightwards double arrow space space space space straight d subscript 1 space equals space 7
From (5),      112 plus straight d subscript 2 space equals space 105 space space space space space space space rightwards double arrow space space space space space space straight d subscript 2 space equals space minus 7
From (1), 28 minus 35 space minus straight d subscript 3 space equals 0 space space space space space space space space space space space space space space space space rightwards double arrow space space space straight d subscript 3 space equals space minus 7
therefore space space space space straight d with rightwards arrow on top space equals 7 straight i with hat on top space minus space 7 straight j with hat on top space minus space 7 straight k with hat on top