Question

Let `veca = 4hati + 5hatj - hatk`, `vecb  = hati - 4hatj + 5hatk` and `vecc = 3hati + hatj - hatk`. Find a vector `vecd` which is perpendicular to both `vecc` and `vecb and vecd.veca = 21`

Answer 1

Let `vecd = xhati + yhatj + zhatk`

Since `vecd` is perpendicular to both `vecc` and `vecb`, so

`vecd.vecc  = 0` and `vecd.vecb = 0`

3x + y - z = 0 ....1

x - 4y + 5z = 0  ....2

`vecd.veca = 21` 

4x + 5x - z = 21 ....3

Solving 1, 2 and 3

`x = (-1/3), y = 16/3 ,  z = 13/3`

`vecd = 1/3 (-hati + 16hatj + 13hatk)`