Distances between planes (intro lin alg class)

In summary, the distance between the planes given by the equations is (1/3, 0, 0) for plane 2 and (√6)/(9) for plane 1.
  • #1
Square1
143
1

Homework Statement



Im working currently with vectors. The question asks for the distance between two planes given by the two following equations:
x + y -2z = 0
3x + 3y -6z = 1

Homework Equations



I know the planes, H1, and H2 are parallel, so I can pick any random point on either, call the point Q, use the other plane as my H, and apply the following formula:

distance(H,Q) = (q-p) dot (n/|n|)



The Attempt at a Solution



Im assuming that someone familiar with vectors and lin. alg is going to answer this so I am not writing the whole essay. But yea if more explanation is needed then let me know.

For H I chose the first plane, so that n from plane 1 is (1, 1, -2). Correct me if I am wrong but I understand that you do need to pick the appropriate plane/line to use the right n? Or can it be either?
For Q on plane 2 I did - (1/3, 0, 0)
For a point P on plane 1 I chose (0,0,0)

After using the corresponding position vectors for the two points, I plug them in the formula and get the result:

(1)/(3√6), but my answers say it is (√6)/(9)

Thanks all.
 
Physics news on Phys.org
  • #2
Square1 said:
distance(H,Q) = (q-p) dot (n/|n|)

Correct

Square1 said:
For H I chose the first plane, so that n from plane 1 is (1, 1, -2).

Correct

Square1 said:
For Q on plane 2 I did - (1/3, 0, 0)
For a point P on plane 1 I chose (0,0,0)
You mean (1/3, 0,0,) for Q, do you?

Square1 said:
After using the corresponding position vectors for the two points, I plug them in the formula and get the result:

(1)/(3√6), but my answers say it is (√6)/(9)

Thanks all.

The correct answer is 1/(3√6)

ehild
 
  • #3
ehild said:
You mean (1/3, 0,0,) for Q, do you?

Yea. I use (1/3, 0, 0) as my Q, and purposefully taking it from plane 2.

Would you mind following up on the other question I snuck in there as well? About picking correct "n"?
 
  • #4
Square1 said:
Would you mind following up on the other question I snuck in there as well? About picking correct "n"?

Yes, of course. Show the problem.

ehild
 
  • #5
Square1 said:
For H I chose the first plane, so that n from plane 1 is (1, 1, -2). Correct me if I am wrong but I understand that you do need to pick the appropriate plane/line to use the right n? Or can it be either?

Ok so that is where the question originated but I will rephrase it...

When picking "n", if the lines are parallel, I can pick either plane 1 or plane 2 to extract "n" right? Or is there any sort of decision I need to make?

Furthermore, if I were to pick the second plane 3x + 3y -6z = 1, my "n" from this plane could be any multiple of that expression, which is why I could say n = (1,1,-2) or n = (3, 3, -6).

Thanks.
 
  • #6
The equation of a plane comes from that any vector in the plane is perpendicular to the normal vector, so their dot product is zero: (r-p)n=0. In components: xnx+yny+znz=pxnx+pyny+pznz=constant

The magnitude of the normal vector does not influence the zero dot product. If it is (1,1,-2) , you equally can choose any vector which is a multiple of this. So (3,3,-6) is also a normal vector, or (-1,-1,2)...Often we choose a unit normal vector, with the components divided by the magnitude.

In the problem, you had two planes. And you found that the normal vectors were multiples of each other. If the normal vectors are of identical direction, the planes are parallel.

ehild
 
  • #7
Thanks so much!
 
  • #8
Square1 said:
Thanks so much!

You are welcome:smile:

ehild
 

1. What are planes in linear algebra?

Planes in linear algebra are two-dimensional surfaces that extend infinitely in all directions. They are represented by equations in the form of ax + by + cz = d, where a, b, and c are constants and x, y, and z are variables.

2. How do you find the distance between two planes?

The distance between two parallel planes can be found by finding the shortest distance between any two points on the planes. This can be done by taking the dot product of the normal vectors of the planes and dividing it by the magnitude of the normal vector. The resulting value is the distance between the planes.

3. Can the distance between two planes be negative?

No, the distance between two planes is always a positive value. It represents the shortest distance between any two points on the planes, and distance is never negative.

4. What is the significance of finding the distance between two planes?

The distance between two planes is important in many applications of linear algebra, such as finding the shortest distance between a point and a plane or determining if a point lies on a certain side of a plane. It is also used in calculating the angle between two intersecting planes.

5. How does the distance between two planes change if one plane is translated?

If one plane is translated, the distance between the two planes will not change as long as the direction and orientation of the planes remain the same. This is because translation only affects the position of the planes in space, not their relationship to each other.

Similar threads

  • Precalculus Mathematics Homework Help
Replies
1
Views
627
  • Precalculus Mathematics Homework Help
Replies
8
Views
1K
  • Precalculus Mathematics Homework Help
Replies
2
Views
1K
  • Precalculus Mathematics Homework Help
Replies
7
Views
2K
  • Precalculus Mathematics Homework Help
Replies
8
Views
2K
  • Precalculus Mathematics Homework Help
Replies
2
Views
1K
  • Precalculus Mathematics Homework Help
Replies
4
Views
1K
  • Precalculus Mathematics Homework Help
Replies
10
Views
2K
  • Precalculus Mathematics Homework Help
Replies
2
Views
1K
  • Precalculus Mathematics Homework Help
Replies
2
Views
940
Back
Top