Calculate missing vector based on the angle I'm supposed to get

Question

Let's say I have a vector (-1,2,-3) and I want to combine this with an other vector, so the angle of these 2 vectors would come out to 30°. How would I go about calculating this? My professor showed us an example for this kind of task but didn't really go into the specifics, just the theory, so I know I have to somehow use the dot product? Not sure how exactly though.

I hope I've managed to explain this well enough, English isn't my first language. Thank you for every answer that could steer me into the right direction with this task.

Answer 1

I know how I'd do it though I don't know if it's how your professor means.

1. take the cross product of this vector with some other vector, preferably a unit vector in one axis to make your life easier. (in other dimensions, you can use the gram-schmidt process instead but that's harder to explain)

$$ v = (-1, 2, -3) \times (1,0,0) = (0,-3,-2) $$

This vector is known to be perpendicular to your input vector.

2. scale both of these vectors to unit vectors, so they have known lengths.

$$\begin {align} \hat u &= \frac{u}{|u|}=\frac{(-1,2,-3)}{\sqrt{14}}\\ \hat v &= \frac{v}{|v|}=\frac{(0,-3,-2)}{\sqrt{13}} \end{align}$$

3. add them together using the sine and cosine of the desired angle to get the final result

$$\begin{align} \hat q &= \hat u \cos\theta + \hat v \sin\theta\\ &= \frac{(-1,2,-3)}{\sqrt{14}}\cdot \frac{\sqrt{3}}{2} + \frac{(0,-3,-2)}{\sqrt{13}}\cdot\frac{1}{2}\\ &= \sqrt{\frac{3}{56}}(-1,2,-3)+\sqrt{\frac{1}{52}}(0,-3,-2) \end{align}$$

Which is a miserable enough pile of numbers that I'm not going to do the addition out, but it is a unit vector and it is indeed $30^\circ$ away from the input vector.