Vectorial geometry, linear algebra (planes, normals, distances)

In summary: I was thinking of doing thisfirst finding the equation for the distance from a point to a planed = |ax1 + bx2 + cx3 + d| / sqrt(a^2 + b^2 + c^2)then finding the distance from a point to a line, and I'm not sure how to do that quite yetthen using the pythagorean theorem to find the distance from r to the line through p and qbut i don't think it's the most efficient wayEDIT: I found this formula tood = ||r x (p - q)|| / ||p - q||which is the magnitude of the cross product of r
  • #1
Flowergirl
3
0

Homework Statement


I am given the following vectors :


Code:
p =   3       q =  2        r =  5
      2            4             3
     -4           -3            -1

They ask to find these:
1. a normal to the plane containing p, q and r.
2. the distance from the origin to the plane containing p, q and r
3. The distance from r to the line containing p and q
4. The matrix and translation of the affine transformation T:ℝ3 → ℝ3 which projects points orthogonally onto the plane containing p, q and r.


Homework Equations



equation for a normal to a plane:
f(x,y,z) = ax + by + cz + d = 0

projection equation:
projvn = (|v . n|/|v|2) . v


The Attempt at a Solution



I will have to edit this in since i need to leave quickly but I would still like to throw this out there, I know the equations, I know the normal is a vector which is orthogonal to the plane and thus who's dot product is equal to zero, but I do not know how to go about it.

As for #2 I do not know how to solve for an equation in which all vectors equal zero.

#3 I think the line containing p and q will be p - q = s, and i must project r onto s to find the distance

for #4 I do not understand what they are asking.
 
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  • #2
Welcome to PF, Flowergirl! :smile:

Let's start with #1.

To find a normal vector, first you need to find 2 vectors that lie within the plane.
And to find a vector in the plane you need to subtract 2 vectors that have their end point in the plane.
Can you find 2 vectors that lie within the plane?
 
  • #3
Thank you for the welcome :)

lets say p - q = n and q - r = m

n = (1, -2, -7)

m = (-3, 1, -4)
You said, originally,
I am given the following vectors :

p = 3 q = 2 r = 5
2 4 3
-4 -3 -1
They ask to find these:
1. a normal to the plane containing p, q and r.
So any two of p, q, and r are in the plane. Of course, any linear combination is then in the plane but it is not necessary to find new vectors in the plane.
 
Last edited by a moderator:
  • #4
Aha! It looks like you already know more or less what you need to do.

Now let's see... when I calculate p-q and q-r I get different vectors.
They are almost the same...
Is it possible you made a couple of mistakes with + and -?

Then you deduce a couple of equations from your matrix.
How did you arrive at those equations?
There appears to be a mistake in that step, since your final vector is not normal to your vectors n and m.
Can you check the dot product of your final vector with your vector m respectively your vector n?
 
  • #5
oh oops, well i redid it and got (-1,-1,1) normal vector but .. how do I express it in equation form? like ax1 + bx2 + cx3 + d = n

EDIT: sorry, I'm retarded, it'd be -x1 - x2 + x3 + 9 = n

for b I have to find a vector from the plane to the origin? or do i just project to point zero... I'm really not quite sure how to go about this

I searched and found that I have to project a vector from the plane onto n, and that will give the distance between the origin and the plane.

I ended up finding d = ||projqn|| = |nTq| / ||n|| = |-9|/(3)1/2
 
Last edited:
  • #6
Looking good. :)

Btw, that "n" in your equation should be zero.

The dot product is effectively a projection.
If you take the dot product of a vector in the plane with the normal vector you found, and divide by the length of the normal vector, you get the distance to the origin.
So yes, you got the right distance.

For #3, to find the distance from r to the line through p and q, you will need a different formula than the ones you've shown so far.
Do you have more formulas?
Or are you supposed to construct one yourself?
 
Last edited:

1. What is the difference between a vector and a point in vectorial geometry?

In vectorial geometry, a point is a specific location in space, represented by its coordinates. A vector, on the other hand, has both magnitude and direction, and is often used to represent a displacement or change in position. In other words, a point is a fixed location, while a vector is a direction and distance from that location.

2. How is linear algebra used in real-world applications?

Linear algebra is used in a variety of fields, including engineering, physics, economics, and computer science. It can be used to solve systems of equations, model and analyze data, and make predictions. For example, it can be used to calculate optimal routes in transportation, forecast stock market trends, and design efficient structures.

3. What is a normal vector and how is it used in vectorial geometry?

A normal vector is a vector that is perpendicular to a given surface or plane. In vectorial geometry, it is often used to determine the angle between two planes, or the shortest distance from a point to a plane. It is also used in computer graphics to determine the orientation of surfaces and to create realistic lighting effects.

4. How are matrices and vectors related in linear algebra?

A matrix is a rectangular array of numbers, while a vector is a one-dimensional array. In linear algebra, matrices can be used to represent linear transformations, which are operations that map vectors from one space to another. Vectors can also be multiplied by matrices to produce new vectors, and can be used to solve systems of linear equations.

5. Can vectorial geometry and linear algebra be used to solve real-world problems?

Yes, vectorial geometry and linear algebra provide powerful tools for solving a wide range of real-world problems, from calculating optimal paths and predicting outcomes, to designing efficient systems and analyzing data. These concepts are used extensively in many fields, including engineering, physics, computer science, and economics.

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