Question

How do you find the vector perpendicular to the plane containing (0,-2,2), (1,2,-3), and (4,0,-1)?

Answer 1

Make two vectors from the 3 points and then take the cross product:

`barv xx baru = 2hati + 17hatj + 14hatk`

Explanation:

Use the 3 points to create two vectors in the plane:

let `baru = (1 - 0)hati + (2 - -2)hatj + (-3 - 2)hatk`

`baru = 1hati + 4hatj - 5hatk`

let `barv = (4 - 0)hati + (0 - -2)hatj + (-1 - 2)hatk`

`barv = 4hati + 2hatj - 3hatk`

Use the cross product to find a vector perpendicular to these two vectors:

`baru xx barv = |(hati,hatj,hatk,hati,hatj),(1,4,-5,1,4),(4,2,-3,4,2)|`

`baru xx barv = hati{(4)(-3) - (-5)(-2)} + hatj{(-5)(4) - (1)(-3)} + hatk{(1)(2) - (4)(4)}`

`baru xx barv = -2hati - 17hatj - 14hatk`

The negative of this is also valid:

`barv xx baru = 2hati + 17hatj + 14hatk`