# Find Value of \alpha for Perpendicular Vector to Plane

• milan666
In summary: The alphas are different because the vector perpendicular to the plane is also perpendicular to the 2 vectors that are parallel to the plane; alpha is the coefficient of the vector perpendicular to the plane. The method you used is alright.

## Homework Statement

I'm given a vector (x,y,$$\alpha$$) (cant state exact number cause of plagarism). I have to find the value of $$\alpha$$ which make the vector perpendicular to a plane. I also have 2 other vectors that are parallel to the plane.

dot product.

## The Attempt at a Solution

I found the value of $$\alpha$$ that is perpendicular to the 2 line parallel to the plane, but it is different from the value i get if i calculate the vector to be perpendicular to the plane itself. Infact, the $$\alpha$$ value perpendicular to the plane is a multiple of the $$\alpha$$ value perpendicular to the 2 vectors parallel to the plane, and its also a different sign. I want to know what is the reason for this and am i doing something wrong? shouldn't i get the same value since the 2 vectors and the plane are parallel?

A vector that is perpendicular to a plane is also perpendicular to all lines that are parallel to the plane.
If the two lines do not have the same direction, a vector perpendicular to two lines that are parallel to a plane, is also perpendicular to the plane.
I can't see what you did wrong without any calculation.

^^^ Agreed. How did you calculate the value of alpha, cross product? Also, let us know the plane you are trying to find it to be perpendicular to.

Thanks for the reply. I would give you the numbers but as i said, it would be plagarism since i am at uni. I used the dot product to find alpha, since cos90=0, the dot product should be zero? And the value of alpha for the plane is twice the value i got for the 2 lines, and is a different sign. Does that maybe mean that its goes in the opposite direction?

It's really hard for me to tell what's going on as it is. Could you just use some different numbers from the ones you are given if you're really worried about plagirism, and show us your problem and calculations that way?

Your professor should be okay with getting help on homework. If this is something more serious then we shouldn't be helping you.

ok ill just use totally different numbers, (the answers are random, they wouldn't actually be that way) and I am not asking for the answer, just an explanation. Suppose the equation of the plane is 3x + 3y + 3z = 7 and the 2 vectors are (1,2,3) and (4,5,6). I need to find the coresponding value of alpha for which the vectors (8,9,9+alpha) is perpendicular to the plane. Let's say when i used the dot product on the plane --- (8,9,9+alpha).(3,3,3) = 0, i got alpha is 4.
And when when i used the dot product with the 2 vectors i got alpha is -2.
Why are the alphas different? shouldn't they be the same? also is the method i used (dot product) alright?

## 1. What is the formula for finding the value of α for a perpendicular vector to a plane?

The formula for finding the value of α is: α = cos^-1 (n • v / |n||v|), where n is the normal vector of the plane and v is the vector that is perpendicular to the plane.

## 2. How do you determine the value of α for a perpendicular vector to a plane using coordinates?

To determine the value of α using coordinates, you can use the dot product formula: α = cos^-1 ((ax + by + cz) / √(a^2 + b^2 + c^2) * √(x^2 + y^2 + z^2)), where (a,b,c) are the components of the normal vector and (x,y,z) are the components of the perpendicular vector.

## 3. Can the value of α be negative or greater than 90 degrees?

Yes, the value of α can be negative or greater than 90 degrees. A negative value indicates that the perpendicular vector is in the opposite direction of the normal vector, and a value greater than 90 degrees indicates that the perpendicular vector is facing away from the plane.

## 4. How does the value of α affect the orientation of the perpendicular vector to the plane?

The value of α determines the angle at which the perpendicular vector is oriented relative to the plane. A value of 0 degrees indicates that the vector is parallel to the plane, and a value of 90 degrees indicates that the vector is perpendicular to the plane.

## 5. Is it possible for a plane to have multiple perpendicular vectors with different values of α?

Yes, it is possible for a plane to have multiple perpendicular vectors with different values of α. As long as the dot product of the normal vector and the perpendicular vector is 0, the vector will be perpendicular to the plane. Therefore, there can be infinitely many perpendicular vectors with different values of α for a single plane.