# Normal plane

1. Homework Statement

Find the normal vector equation (the equation with red in the picture below) of the plane which is normal on the vector n=(2,-2,1), and it is on distance 5 from O (i.e p=5)

2. Homework Equations

On this image are all the relevant equations.
http://img232.imageshack.us/img232/6030/28762075sm2.jpg [Broken]

3. The Attempt at a Solution

I should find $$n_o$$, and then substitute in the equation.
$$n_o=\frac{n}{|n|}$$
So n=2i-2j+k, |n|=3, p=5
$$n_o=\frac{2i-2j+k}{3}$$

So the equation:

$$r * (\frac{2i-2j+k}{3}) - 5 =0$$

The problem is that in my book they don't divide it by 3? Why ? Is my way correct?

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Hootenanny
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Does it ask you for the unit normal, or just the normal?

The angle between the plane and the vector is 90 degrees. Nothing more.

Hootenanny
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The angle between the plane and the vector is 90 degrees. Nothing more.
No, my point is the question asks you to find the normal vector, not the unit normal. There is no need to divide by it's magnitude.

Actually, we should find the unit normal. I think we should divide it by its magnitude, because the definition says $$|n_o|=1$$

Hootenanny
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In the question you stated,
Find the normal vector equation (the equation with red in the picture below) of the plane which is normal on the vector n=(2,-2,1), and it is on distance 5 from O (i.e p=5)
there is no mention of a unit normal. Hence the question wants n not n0.

But the formula is with $$n_o$$
http://img232.imageshack.us/img232/6030/28762075sm2.jpg [Broken]

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Hootenanny
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Yes but where is that image taken from, a course text, lecture notes?

Yes but where is that image taken from, a course text, lecture notes?
from my textbook. It also says that $|n_o|=1$

Hootenanny
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Does it occur to you that you are not meant to use that equation in this question?

Does it occur to you that you are not meant to use that equation in this question?
Then, when I should use this equation?

Hootenanny
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Perhaps,

$$r\cdot \boldmath{n} - p = 0$$

Since you are not asked for the unit normal, simply the normal vector.

Perhaps,

$$r\cdot \boldmath{n} - p = 0$$

Since you are not asked for the unit normal, simply the normal vector.
If we write
$$r\cdot \boldmath{n} - p = 0$$
then it wouldn't be plane, because equation of plane is only when $|n_o|=1$.

Hootenanny
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If we write
$$r\cdot \boldmath{n} - p = 0$$
then it wouldn't be plane, because equation of plane is only when $|n_o|=1$.
Yes it is still an equation of a plane, the very same plane in fact! The equation of a plane can be derived from any normal vector, not just a normalised one!

Now, I even doubt why p=5, it should be p=3, since OP=p$n_o$. Well, I think I am tottaly confused about the whole thing...

Hootenanny
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Would it be possible for you to scan the actual diagram from your textbook and post it here?

Here are the pictures of the lection:
http://pic.mkd.net/images/138086IMG_1228.jpg [Broken]
http://pic.mkd.net/images/843151IMG_1229.jpg [Broken]
http://pic.mkd.net/images/709895IMG_1230.jpg [Broken]

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Hootenanny
Staff Emeritus
Gold Member
Thanks for the pictures, could you also type out the question in full, with all the information given?

Find the normal vector equation of the plane which is normal on the vector n=(2,-2,1), at distance 5 from (0,0,0) (the coordinate start, sorry if I mistranslated). You can see the task on the third picture 88 page/1 task.

Hootenanny
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Thanks, perhaps something is getting lost in translation here. Does the vector n=(2,-2,1) lie on the plane, i.e. is it part of the plane, or is it perpendicular to it?

n is perpendecular to the plane... As you can see it have 2 rows of text. That's the full text.

Hootenanny
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Ahh, I follow now. The diagram is somewhat confusing, however, it should become clear if you note that

$$\vec{OP}\neq \vec{n_0}$$

Although it appears so in the diagram, it is not the case.

In this case it is not. But how will I find the equation then?

Hootenanny
Staff Emeritus
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In this case it is not. But how will I find the equation then?
As I said before,

$$\underline{r}\cdot\underline{n} - p = 0$$

You are given both n and p.

As I said before,

$$\underline{r}\cdot\underline{n} - p = 0$$

You are given both n and p.
can u tell me please where did u find this equation
$$\underline{r}\cdot\underline{n} - p = 0$$ from?