calculate vector after rotating it towards another by angle θ in 3D space

Question

Unit vectors V and D lie on a 3D space. They have the same starting point. I want to rotate the vector V towards to vector D, but only by angle θ.

Given that we know:

1. Vectors V and D are unit vectors.
2. We know V(x,y,z) and D(x,y,z).
3. We know the angle between the two vectors V and D, Δφ.
4. We also know the angle θ, where the vector V will rotate to approach vector D.
5. We know the starting point "O" where the three vectors start from.

We now want to calculate vector Z, which will also be a unit vector. Is it possible to calculate the coordinates of vector Z, with this information given above?

Do you have any ideas how this problem can be solved?

Answer 1

What you need to do is to define a third axis in the plane of the V and D vectors such that a 90 degree rotation from V points in this direction. I'll call the unit vector that points in this direction D'. With this, your Z vector is easy:

Z = cos(theta)*V + sin(theta)*D_tick;

So how to compute D'? That, too is easy. First compute a vector orthogonal to V and D using the cross product. Call this W: W = V×D. Next compute a vector orthogonal to W and V: D' = W×V = (V×D)×V. This points in the right direction, but it will only be a unit vector if your V and D are orthogonal. So normalize: D' = D'/||D'||, where ||D'|| is the magnitude of the vector D'. If you have a vector math package, you can do this via

D_tick = ((V.cross(D)).cross(V)).normalize();

One caveat: What if ||D'|| is zero? That happens if and only if your Δφ is a multiple of pi radians (or 180 degrees). Alternatively, it happens when V and D are parallel or anti-parallel to one another. Your question is ill posed in this special case. You should check for this special case.

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Addendum
My (V×D)×V and comingstorm's D - V*((D·V)/(V·V)) are one and the same for vectors in three dimensional space. Because V is a unit vector, his D-V*((D·V)/(V·V)) reduces to D-V(D·V). My (V×D)×V is equal to D(V·V)-V(D·V) per the vector triple product identity (http://mathworld.wolfram.com/VectorTripleProduct.html), and this reduces to D-V(D·V), again because V is a unit vector.

Answer 2

One way to do this is to find the component of D perpendicular to V, scale it up to equivalent length, and do a vector sum using sin() and cos():

D_perp = D - V * ((D . V)/(V . V))

D_perp_scaled = D_perp * (|V|/|D_perp|)

result = cos(theta) * V + sin(theta) * D_perp_scaled

This is well-defined unless D is parallel to V, which will make |D_perp| == 0 and cause problems with the division. This isn't really surprising: in that case your plane of rotation is ill-defined -- it isn't clear which direction you should rotate!

Mathematically, this method for finding the perpendicular is equivalent to the cross-product method cross(cross(V,D),V) mentioned in other answers, but is perhaps a bit simpler, and works for any vector space, (e.g., 2-D and 4-D vectors, not just 3-D).