For vector AB, how to find vector B given vector A and the angle between

In summary: This will give you a new vector A which is (x_0, y_0, 1), and is at the same length as vector A, but has the same angle as vector A.
  • #1
aamirnshah
4
0
Hello,

I have a problem I've been trying to solve. I originally have vector AB. I know both vector A, vector B, and the angle between. They are on the same plane.

Now say I change the angle between the two vectors. They still remain the same magnitudes, and remain on the same plane. Assume vector A is fixed. How can I find the new updated vector B.

My attempt at a solution:

|A||B|cos theta = A*B

A, |B|, and theta are known

Equation 1)
|A||B|cos theta = axbx + ayby + azbz

Equation 2)
ax + by + cz = d
This is the equation of the plane that the vectors are located on.

My problem is I need one more equation to solve this. Maybe I am going about this the wrong way? Any help is appreciated.

Thanks,
AS
 
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  • #2
Whoa, hold up.

"I originally have vector AB."

I'll assume that, by this, you mean you know vector A, and you know vector B, in some sort of coordinate system... say, polar, for convenience.

It just so happens that the vectors have an angle between them which can be found by subtracting angle_B - angle_A = angle_difference.

Then if the angle is changed so that angle_difference' = angle_difference + angle_delta, then angle_B' = angle_B + angle_delta... no problem.

If you have the vectors in cartesian form... well, convert to polar, then convert back when you're done using the above method.

If you don't originally know angle_A, then just call angle_A = 0, and go from there.

Maybe I'm just missing some subtlety here...
 
  • #3
Hi,
First, you have to convert your second equation into a an equation with bx, by, bz.

For your third equation how about one saying that the length of the new vector B is the same as the length as the original B.


Another way you might think about the problem is to find the direction about which you want to rotate B. If you want to keep B coplanar with A then rotate B about the axis perpendicular to both A and B.

Then the wikipedia page "Rotation Matrix" has the explicit matrix which will rotate a vector by any angle about an arbitray axis. It's a horrible looking matrix multiplication but you would just be plugging in numbers and doing arithmetic rather than solving a system of equations.
 
  • #4
alexgs said:
Hi,
First, you have to convert your second equation into a an equation with bx, by, bz.

If I took 3 points on the plane, and obtained some formula ax + by + cz + d = 0 with the normal vector (a,b,c), isn't this sufficient enough for the second equation?

alexgs said:
For your third equation how about one saying that the length of the new vector B is the same as the length as the original B.

I tried this. I end up with a complex third equation that looks something like this :

some scalar = x^2 + x + y^2 + y + z^2 + z.

I am unsure how to use this equation with the other 2.

alexgs said:
Another way you might think about the problem is to find the direction about which you want to rotate B. If you want to keep B coplanar with A then rotate B about the axis perpendicular to both A and B.

This method might work. I will try it. Thanks a lot!
 
  • #5
aamirnshah said:
Hello,

I have a problem I've been trying to solve. I originally have vector AB. I know both vector A, vector B, and the angle between. They are on the same plane.

Now say I change the angle between the two vectors. They still remain the same magnitudes, and remain on the same plane. Assume vector A is fixed. How can I find the new updated vector B.

Thanks,
AS


I think the solution won't be unique. Imagine the plane to be z=1 and the point A is (0,0,1). Given an angle \theta we want to look for the point (x,y,1) such that
[tex]\tan^2 \theta = x^2 + y^2[/tex].
There will be infinite solutions.
 
  • #6
I assume you have some vector A, which, in a given coordinate system, can be represented as [itex]\left(x_0, y_0)\right)[/itex], and want to find vector A, having the same length as A but at angle [itex]\theta[/itex] to A.

Let [itex]\phi[/itex] be the angle A makes with the x-axis. Then, assuming that [itex]\theta[/itex] is measured counterclockwise from A, B makes angle [itex]\theta+ \phi[/itex] with the x-axis. Its components are [itex](|A|cos(\theta+ \phi),|A|sin(\theta+ \phi)[/itex].

Now use the facts that [itex]cos(\theta+ \phi)= cos(\theta)\cos(\phi)- sin(\theta)sin(\phi)[/itex], [itex]sin(\theta+ \phi)= sin(\theta)cos(\phi)+ cos(\theta)sin(\phi)[/itex], [itex]cos(\phi)= x_0/|A|[/itex], and [itex]sin(\phi)= y_0/|A|[/itex].
 

1. How do I find the magnitude of vector B?

To find the magnitude of vector B, you can use the formula |B| = |A| * sin(angle between A and B). This formula uses the magnitude of vector A and the sine of the angle between vector A and B to determine the magnitude of vector B.

2. Can I use the cosine of the angle between A and B to find the magnitude of vector B?

No, the cosine of the angle between A and B is used to find the scalar projection of vector B onto vector A, not the magnitude of vector B.

3. How do I find the direction of vector B?

The direction of vector B can be determined by using the tangent of the angle between A and B. The tangent will give you the ratio of the opposite and adjacent sides of the triangle formed by vector A and B, which can be used to determine the direction of vector B.

4. What if I don't know the angle between A and B?

If you do not know the angle between A and B, you can use the dot product of vector A and B to find the cosine of the angle. Then, you can use the inverse cosine function to find the angle between A and B, and use that angle in the formula to find the magnitude of vector B.

5. Can I use this method to find the magnitude and direction of any vector?

Yes, this method can be used to find the magnitude and direction of any vector as long as you know the magnitude of one vector and the angle between the two vectors. However, it is important to note that this method only works for two-dimensional vectors.

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