Question

Find a unit vector perpendicular to each of the vector  `veca  + vecb` and `veca - vecb`, where `veca = 3hati + 2hatj + 2hatk` and `vecb = hati + 2hatj  - 2hatk`.

Answer 1

`veca = 3hati + 2hatj + 2hatk, vecb = hati + 2hatj - 2hatk` 

`veca + vecb = 4hati + 4hatj, veca + vecb = 2hati + 4hatk`

`(veca + vecb) xx (veca + vecb) = |(hati, hatj, hatk), (4, 4, 0), (2, 0, 4)|`

`= (16 - 0)hati - (16 - 0)hatj + (0 - 8)hatk`

`= 16hati - 16hatj - 8hatk`

∴ Unit vector perpendicular to both `(veca + vecb)` and ` (veca - vecb)` is given by

`= pm ((veca + vecb) xx (veca - vecb))/|(veca + vecb) xx (veca - vecb)|`

`= pm (16 hati - 16hatj - 8hatk)/sqrt((16)^2 + (-16)^2 + (-8)^2)`

`= pm (8 (2hati - 2hatj - hatk))/(8 sqrt (4 + 4 + 1))`

`= pm (2hati - 2hatj - hatk)/3`

`= pm 2/3 hati pm 2/3 pm 1/3 hatk`