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Find a vector perpendicular to the plane containing the given points. (1, 2, 3), (-4,...

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claymore | Student, Undergraduate | eNoter

Posted July 8, 2012 at 12:02 AM via web

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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.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted July 8, 2012 at 10:50 AM (Answer #1)

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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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