How to express a vector B given a known vector A and known angle

Question

1) There are two vectors: vector A and vector B. Both A and B are $1 \times K$ dimension. The angle $\theta$ between A and B is known , how can I get an expression of vector B in terms of vector A and sin($\theta$)?

like: $B=f(A,\text{sin}(\theta))$, Assume A, B are normalized

2) If the answer to the first question is not possible, now assume A and B are both known and angle $\theta$ is known, is it possible to express difference vector: A-B in terms of sin($\theta$)?

Any suggestion will be appreciated! thanks!

Answer 1

the problem is indeterminate in more than two dimensions.

for the 2-D case there are only two possibilities. to see them represent the unit vector $A$ as a complex number $e^{i \phi}$ so $B=e^{\phi \pm \theta}$

Answer 2

In 2D, we can express $B$ as a rotation of $A$ with angle $\theta$ (or $-\theta$): $$B_x = A_x \cos\theta - A_y \sin\theta$$ $$B_y = A_x \sin\theta + A_y\cos\theta$$

In 3D, the problem is underspecified: There are infinitely many candidates for $B$, forming a cone around $A$. You need to add some contraints, such as $A$ and $B$ lying in a certain plane.