How do you find the sine of the angle between two vectors?

Question

I do not know what the sine of the angle between two vectors is. I think it may be the vector created by connecting the tips of the two vectors but I am not sure.

How do you find the sine of the angle between two vectors?

Answer 1

Translate your two vectors so that their tails are at the origin. Then draw a line through each of those two vectors. The smaller of the two angles is the called the "angle between the two vectors".

Hint on how to find it: The angle $\theta$ between two vectors $\vec u$ and $\vec v$ is given by the formula $$\theta = \arccos\left(\frac {\vec u\cdot\vec v}{|\vec u||\vec v|}\right)$$

Or -- here's a slightly harder way (because it involves calculating a cross product). You know that $|\vec u\times \vec v| = |\vec u||\vec v|\sin(\theta)$, so find the cross product and all of the norms and then plug in.

Answer 2

Let $\theta$ be the angle between the two vectors. Now you can use, $$\cos\theta={\vec v\cdot\vec w\over |\vec v||\vec w|}$$ and $$\sin\theta=\sqrt{1-\cos^2\theta}.$$

Answer 3

In general for $\mathbf{a},\mathbf{b}\in\mathbb{R}^3$, we have $$\|\mathbf{a}\times \mathbf{b}\|=\|\mathbf{a}\|\|\mathbf{b}\|\sin\theta,$$ where $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}$.