The dot product

You can add vectors, and that is easy to picture: hang two arrows one after the other. But how do you multiply two arrows? The answer is surprising: the result is not an arrow at all, but a single number. That is why the calculation is called the scalar product, because scalar is simply the word for an ordinary number without direction.

The calculation itself

The calculation is surprisingly simple: multiply the $x$ values, multiply the $y$ values and add the two. For $\vec{a} = (3|4)$ and $\vec{b} = (2|1)$ that gives $3 \cdot 2 + 4 \cdot 1 = 10$. Done. No arrow, no magnitude, just the number 10. For vectors in space you simply add a third term.

Zero means perpendicular

Now comes the reason this calculation is so useful. Take $\vec{a} = (3|0)$, an arrow to the right, and $\vec{b} = (0|5)$, an arrow upwards. The dot product is $3 \cdot 0 + 0 \cdot 5 = 0$. That is no accident: the dot product is zero exactly when the two vectors are perpendicular to each other. So you have a right-angle test that needs no drawing and no protractor.

The angle in between

Perpendicular is only the special case. In general the dot product is linked to the angle: $\vec{a} \cdot \vec{b} = |\vec{a}| \cdot |\vec{b}| \cdot \cos \alpha$. Rearranged for $\cos \alpha$ this gives you the angle between any two vectors without drawing anything. And because $\cos 90° = 0$, the rule from before drops out automatically.

The sign alone already tells you something, before you calculate anything. If the dot product is positive, the two vectors roughly point the same way and the angle is less than 90°. If it is negative, they point away from each other and the angle is more than 90°. If it is 0, they are exactly perpendicular.