How Do You Solve a Plane Equation in R^4 and Find Its Normalized Normal?

[concon]

Homework Statement

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

Homework Equations

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

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?

 

[Mark44]

Homework Statement

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

Homework Equations

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

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

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 = <?, ?, ?, ?>

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

Just put in the coordinates of the normal, and you're done.

Part B

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

There's a formula for the distance between two points in R⁴.

 

[concon]

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.

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!

 

[Mark44]

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?

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.

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

If A = (a₁, a₂, a₃, a₄) and B = (b₁, b₂, b₃, b₄) are two points in R⁴, then $d(A, B) = \sqrt{ (a_1 - b_1)^2 + (a_2 - b_2)^2 + (a_3 - b_3)^2 + (a_4 - b_4)^2}$

 

[concon]

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 $d(A, B) = \sqrt{ (a_1 - b_1)^2 + (a_2 - b_2)^2 + (a_3 - b_3)^2 + (a_4 - b_4)^2}$

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

 

[Mark44]

Okay so H and H2 have same norm of (4,4,5,-6)?

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.

What then is the "normalized norm n of the plane"?

Make that "normalized normal." A normalized vector is one whose length is 1.