Find normalized normal of plane H parallel to H2

In summary, the task at hand is to find the equation of a plane H in R^4 that contains the point P = (2,-1,10,6) and is parallel to plane H2: 4a + 4b + 5c - 6d = 3. To accomplish this, the normalized normal of plane H must be found, which has an angle theta with the normal n = (4,4,5,-6) of H2 such that cos(theta) > 0. The distance from the point (2,2,-1,-2) to the plane H also needs to be calculated. To find the equation for plane H, the formula 0 = n1(a
  • #1
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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?
 
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  • #2
I know that I can find a point on the plane H that is closest to the given point (2,2,-1,-2), but I am unsure how to calculate the distance
 

1. What is a normalized normal vector of a plane?

A normalized normal vector of a plane is a vector that is perpendicular to the plane and has a length of 1. It is often used in mathematical calculations and graphics to determine the orientation of the plane.

2. How do you find the normalized normal vector of a plane?

To find the normalized normal vector of a plane, you can use the cross product of two non-parallel vectors in the plane. Then, divide the resulting vector by its magnitude to get a vector with a length of 1.

3. What does it mean for a plane to be parallel to another plane?

Two planes are parallel if they never intersect or touch each other, even when extended infinitely. This means that they have the same slope or direction in 3-dimensional space.

4. How is the normalized normal vector of a plane related to its parallel plane?

The normalized normal vector of a plane is perpendicular to the plane and can be used to determine the orientation of the plane. If two planes are parallel, then their normalized normal vectors will be parallel as well.

5. Can the normalized normal vector of a plane change if the plane is rotated?

Yes, the normalized normal vector of a plane will change if the plane is rotated. This is because the orientation of the plane has changed, and therefore the vector perpendicular to it will also change.

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