Finding an angle between two vectors without a calculator

Question

I have an exercise where I have to find the angle between 2 vectors without using the calculator. I know the formula which gives you the cosine of the angle between the 2 vectors, but I don't know how to derive the angle from the cosine of the angle manually.

This is the question:

Find the angle $\alpha$ (from the cosine) between these pairs of vectors (no calculator).

This are the pair of vectors:

$$u = (\sqrt{3}, \sqrt{2}), v = (-\sqrt{2}, \sqrt{3})$$ $$u = (\sqrt{3}, \sqrt{3}), v = (\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}})$$ $$u = (1, 0), v = (\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}})$$

To find the cosine for the first pair I used the following method:

1. Find $u \cdot v$
2. Find $||u||$ and $||v||$
3. Solve the expression $\frac{u \cdot v}{||u|| \cdot ||v||}$

The 3. point gives me the cosine of the angle between $u$ and $v$, but what about finding the angle (without using the calculator), say $\alpha$?

Answer 1

You know (or should know) the cosines of a variety of angles such as $0$, $\frac{\pi}6$, $\frac{\pi}4$, $\frac{\pi}3$, $\frac{\pi}2$, $\pi$, and so on. (Or, if you prefer degrees, $0°$, $30°$, $45°$, $60°$, $90°$, $180°$, and so on.)

Each those pairs of vectors gives a cosine or its negative in that list. So you can do these problems they way you outline.

Another way, however, is to do this geometrically. Sketch a coordinate system and the two vectors. The angle should then be obvious. (I was able to do each problem in my head using this technique.)

Let us know if you cannot figure out a particular vector.

Answer 2

In all three cases, the vectors are chosen to give values of $\cos \theta$ you are expected to know. If you are working without a calculator, you should expect that.