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I want to find the projection of the point $M(10,-12,12)$ on the plane $2x-3y+4z-17=0$. The normal of the plane is $N(2,-3,4)$.

Do I need to use Gram–Schmidt process? If yes, is this the right formula?

$$\frac{N\cdot M}{|N\cdot N|} \cdot N$$

What will the result be, vector or scalar?

Thanks!

David
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Ofir Attia
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  • Isn't GS process used to build an orthonormal basis in an inner producet vector space out of any given basis? – DonAntonio Jul 25 '13 at 08:18
  • but it use to find the projection of each vector in the basis so if I will do that on 1 vector it will give me the projection. this is what I assume, there is another formula for it? what about $\frac{M \cdot N}{|N|}$? – Ofir Attia Jul 25 '13 at 08:23
  • Ok, so the plan perhaps is to find to lin. independent vectors in $,\Bbb R^2,$ that generate the plane (watch out: the given plane is not a vector subspace) and then to apply GS on these vectors? – DonAntonio Jul 25 '13 at 08:26
  • See this. – Tony Piccolo Jul 25 '13 at 10:04

2 Answers2

1

Set the projection point on the plane as $P=(x,y,z)$.

You need three equations:

  1. Point $P$ on the plane. $$2x-3y+4z=17$$

  2. $\vec{MP}\perp plane$

$$\vec{MP}\perp \vec{PQ_1}$$

$$\vec{MP}\perp \vec{PQ_2}$$

where $Q_1$ and $Q_2$ are two different points on the plane.

Because $\vec{MP}// \vec{N}$, you can use $\vec{N}$ instead of $\vec{MP}$ above.

eccstartup
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0

You can use calculus to minimize the distance (easier: the square of the distance) of M from the generic point of the plane.
Use the equation of the plane to drop a variable, obtaining a function of two independent variables: compute the partial derivatives and find the stationary point. That's all.

Tony Piccolo
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