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

• concon
In summary: You should be able to figure out what it means to normalize a vector.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
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?

concon said:

## 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 = <?, ?, ?, ?>
concon said:
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.
concon said:
Part B

d(x,y) ||y-x||
Does this apply to planes?
There's a formula for the distance between two points in R4.

Mark44 said:
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!

concon said:
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.
concon said:
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}##

Mark44 said:
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"?

concon said:
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.
concon said:
What then is the "normalized norm n of the plane"?
Make that "normalized normal." A normalized vector is one whose length is 1.

## 1. What is linear algebra?

Linear algebra is a branch of mathematics that deals with linear equations, vectors, and matrices. It is used to solve problems involving multiple variables and is commonly used in fields such as physics, engineering, and computer science.

## 2. What are planes in linear algebra?

In linear algebra, a plane is a flat, two-dimensional surface that extends infinitely in all directions. It can be represented by a linear equation with two variables, typically denoted as x and y.

## 3. What are vectors in linear algebra?

A vector in linear algebra is a quantity that has both magnitude and direction. It can be represented as an arrow pointing in a specific direction with a specific length. Vectors are commonly used to represent forces, velocities, and other physical quantities.

## 4. How is linear algebra used in real life?

Linear algebra has many real-life applications, including computer graphics, data analysis, and engineering. It is used to solve problems involving multiple variables and to model real-world situations.

## 5. What are some common operations in linear algebra?

Some common operations in linear algebra include vector addition and subtraction, scalar multiplication, matrix multiplication, and finding the inverse of a matrix. These operations are used to solve systems of linear equations and to manipulate vectors and matrices for various applications.

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