Find a vector perpendicular to the plane containing the given points.

(1, 2, 3), (-4, 2, -1), and (5, -3, 0)

This question is related to perpendicular vectors. I have no idea on how to set up the problem. Any help would be greatly appreciated.

### 1 Answer | Add Yours

You need to remember that the cross product of the vectors formed from the given points represents the orthogonal vector to the plane.

You should form two vector such that:

`bar u = (-4-1)bar i + (2-2) bar j + (-1-3)bar k`

`bar u = -5 bar i - 4 bar k`

`bar v = (5-1)bar i + (-3-2) bar j + (0 - 3)bar k`

`bar v = 4bar i - 5 bar j - 3 bar k`

Using the cross product as normal vector yields:

`bar n = bar u X bar v = [[bar i , bar j , bar k],[-5 , 0 , -4],[4 , -5 , -3]]`

`bar n = 25bar k - 16 bar j - 20bar i - 15 bar j`

`bar n = - 20bar i - 31 bar k + 25 bar k`

**Hence, evaluating the perpendicular vector to the plane containing the given points yields `bar n = - 20bar i - 31 bar k + 25 bar k.` **

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