# Linear algebra, planes, vectors help please

1. Mar 14, 2014

### concon

1. The problem statement, all variables and given/known data

Find equation of plan H in R^4 that contains the point P= (2,-1,10,6)
and is parallel to plain H2: 4a +4b + 5c-6d = 3 then answer the following questions:
A. find normalized normal of plane H which has an angle theta with the normal n= (4,4,5,-6) of H2 such that cos(theta) >0

B.Find the distance from (2,2,-1,-2) to the plane H

2. Relevant equations
0 = n1(a-p1) + n2(b-p2) + n3(c-p3) + n4(d-p4)

3. The attempt at a solution

So for part A:
I know that if they are parallel then the normal of H2 equals to some constant k times normal of H

N2 = kN
and I believe I found equation for H:
0 = n1(a-2) + n2(b+1) + n3(c-10) + n4(d-6)
I just do not know where to go from there

Part B

d(x,y) ||y-x||
Does this apply to planes?

2. Mar 14, 2014

### Staff: Mentor

Since you're given plane H2 as 4a + 4b + 5c - 6d = 3, you should be able to write the coordinates of a normal to this plane by nothing more than inspection.

N = <?, ?, ?, ?>
Just put in the coordinates of the normal, and you're done.
There's a formula for the distance between two points in R4.

3. Mar 14, 2014

### concon

Thank you for your reply! Yes by inspection the Normal for H2 is (4,4,5,-6), but does that mean that H and H2 have the same normal since they are parallel?

Also, what is the equation for distance between two points in R4?
Thanks again!

4. Mar 15, 2014

### Staff: Mentor

Yes, sort of. The same vector will be normal to both planes, but the two vectors don't have to be equal - one could be a scalar multiple of the other.
If A = (a1, a2, a3, a4) and B = (b1, b2, b3, b4) are two points in R4, then $d(A, B) = \sqrt{ (a_1 - b_1)^2 + (a_2 - b_2)^2 + (a_3 - b_3)^2 + (a_4 - b_4)^2}$

5. Mar 15, 2014

### concon

Okay so H and H2 have same norm of (4,4,5,-6)?
What then is the "normalized norm n of the plane"?

6. Mar 15, 2014

### Staff: Mentor

You should not use "norm" to talk about the normal to a plane. The term "norm" is used for something entirely different, and is something like the distance between two things.

I didn't say that H and H2 have the same normal. For example, I can see that the plane x + 2y - 4z = 0 has <1, 2, -4> as a normal. I can also see by inspection, that the plane 2x + 4y - 8z = 3 has <2, 4, -8> as a normal, but the latter vector is just a scalar multiple of the first. Any two vectors that are normal to these planes have to be scalar multiples of each other.
Make that "normalized normal." A normalized vector is one whose length is 1.