Solve for the angle of two straight lines on a 3 dimensional plane.

Question

On a $3$ dimensional plane:

Given that two vectors $\left(x_1,y_1,z_1\right)$ and $\left(x_2,y_2,z_2\right)$ are known and that each of these vectors belong to two separate straight lines intersecting each other at $(0,0,0)$

What formula can be used to calculate the angle formed by the intersecting lines?

Answer 1

If you have two vectors $\mathbf v$ and $\mathbf w$ that both start from the origin, their dot product is $$ \mathbf v\cdot\mathbf w = \|\mathbf v\| \|\mathbf w \| \cos(\theta) $$ where $\theta$ is the angle between them, and $\|\mathbf v\|$ is the magnitude of $\mathbf v$.

Answer 2

You already know they intersect and each belong to a straight line going through $(0,0,0)$, so you can safely assume their cartesian coordinates as originating from $(0,0,0)$.

Now you just need to build the dot product and solve for your angle and you're done!