# Tangent plane, why is it orthongonal and not parallel?

## Homework Statement

Find equation of the plane

The plane though P(6,3,2) and is perpendicular to vector <-2,1,5>

Why would it be -2(x - 6) + (y - 3) + 5(z - 2) = 0?

If it is perpendicular to <-2,1,5>, shouldn't be the cross product of some other vector with this? Using <-2,1,5> wouldn't mean it is parallel to the vector instead?

## The Attempt at a Solution

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Dick
Homework Helper
Are you saying the CROSS product of two perpendicular vectors is zero? If not what are you saying?

I am saying that

[PLAIN]http://img34.imageshack.us/img34/9647/unledseh.jpg [Broken]

A plane's direction is determined by it's normal vector right? So if it is perpendicular to the vector <2,-1,5>, why would its normal vector be the same as <2,-1,5>?

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Dick
Homework Helper
I am saying that

[PLAIN]http://img34.imageshack.us/img34/9647/unledseh.jpg [Broken]

A plane's direction is determined by it's normal vector right? So if it is perpendicular to the vector <2,-1,5>, why would its normal vector be the same as <2,-1,5>?
Perpendicular means the same thing as normal, as far as I know. If the plane is perpendicular to <2,-1,5> then it's also normal to <2,-1,5>. I'm really not sure what your question is. The normal vector is not parallel to the plane if that's what confusing you. Describing it as the 'direction of the plane' doesn't mean it's parallel to the plane.

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But <2,-1,5> is just a scalar multiple of <2,-1,5> (scalar = 1), that is parallel?

Dick
Homework Helper
But <2,-1,5> is just a scalar multiple of <2,-1,5> (scalar = 1), that is parallel?
I may have figured out what is confusing you. A plane doesn't have a unique parallel direction. So the phrase 'direction of the plane' means its normal. Not its parallel. See my previous post.

Is it because of this?

[PLAIN]http://img841.imageshack.us/img841/6863/unleditl.jpg [Broken]

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HallsofIvy