Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

The situation:

In 3D space there are two vectors (A, B) of equal length L, but with different directions.

The beginning points of these vectors are located at a distance of L as well. They could be visualized like this (angles are not equal, both are beginning at the bottom of the image and going up):

A ->  \_|/  <-B

The problem:

I need to determine an angle alpha, by which to rotate vector B around axis Y (Which is given as one of vectors orthogonal to the line between the beginnings of A and B) to reach a situation, where the endings of A and B are also at distance L (Basically to get a 3 dimensional shape from 4 vectors with equal length) If it is hard to calculate an angle, a method for finding coordinates forone of the possible new endpoints for B would also be ok.

The question is:

How can I calculate this angle (one of the two angles), and how can I determine if for given vectors it is even possible to do?

share|improve this question

1 Answer 1

Best I can tell, this is an application of Rodrigues' rotation formula. If you provide the axis of rotation $\hat{Y}$, and angle of rotation $\theta$, then the rotation formula returns a rotated vector: $$\vec{A} = \vec{B}\cos\theta + (\vec{B} \times \hat{Y})\sin\theta + \hat{Y}(\hat{Y}\cdot \vec{B})(1-\cos\theta)$$ Note that depending on your basis vectors and whether you measure the angle clockwise or counterclockwise, the middle term $\vec{B} \times \hat{Y}$ may be reversed: $\hat{Y} \times \vec{B}$. See the Wikipedia article for reference.

Concerning the last difficulty about the tails of $\vec{A}$ and $\vec{B}$ remaining a distance $L$ from each other, if you can additionally provide the direction of $\hat{L}$, then you should be able to apply the rotation formula as if the two vectors originate from the same point and then shift the resultant vector such that $\vec{A}_{\text{shifted}}=(\vec{A_{\text{tip}}}+\vec{L})-(\vec{A_{\text{tail}}}+\vec{L})$, where $\vec{A_{\text{tip}}}$ and $\vec{A_{\text{tail}}}$ are measured relative to the origin.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.