Angle between projected vectors

[emma83]

Homework Statement

Given two 3-vectors U and V with an angle A between them in a plane P, find the plane P' in which the angle between the 2 projected vectors U' and V' is B.
For instance, if you consider the basis vectors X(1,0,0) and Y(0,1,0) in the plane Z=0, find the plane in which the angle between the projected vectors X' and Y' is Pi/4.

Homework Equations

The Attempt at a Solution

I considered the example with the plane P given by x+Ay+Bz+D=0, then expressed the projections X' and Y' on it and with the condition (X'.Y')=X'.Y'.cos(Alpha) and Alpha=Pi/4 I find a complicated relation of degree 4 between A and B. I guess there is an easier way to proceed to find the equation of the plane. Thanks for your help!

 

[mighty2000]

Thank you for your question. This problem can be solved using basic trigonometry and vector operations.

First, let's define the given vectors U and V as 3-vectors in the plane P. We can express them as U = u1X + u2Y + u3Z and V = v1X + v2Y + v3Z, where X, Y, and Z are the basis vectors in the plane P and u1, u2, u3, v1, v2, v3 are the corresponding components of U and V.

Next, we need to find the projections of U and V onto the plane P'. We can do this by taking the dot product of U and V with the normal vector of P', which we can call N. The dot product of two vectors is given by A.B = |A||B|cos(θ), where θ is the angle between the two vectors. In this case, we want the angle between the projected vectors U' and V' to be B, so we can write this as U'.V' = |U'||V'|cos(B).

Now, we can expand the dot product U'.V' using the projections of U and V onto the plane P'. We can express U' and V' as U' = u1'X + u2'Y and V' = v1'X + v2'Y, where u1' and u2' are the projections of u1 and u2 onto the plane P' and v1' and v2' are the projections of v1 and v2 onto the plane P'. Therefore, we have U'.V' = (u1'u1 + u2'u2)X.Y + (u1'v1 + u2'v2)X.Y + (u1'v1 + u2'v2)Y.Z + (u1'v1 + u2'v2)Z.X.

Using the trigonometric identity cos(A+B) = cos(A)cos(B) - sin(A)sin(B), we can rewrite the dot product U'.V' as U'.V' = u1'v1'cos(B) + u2'v2'cos(B) - u1'v2'sin(B) - u2'v1'sin(B).

Now, we can equate this to the dot product