The dot product multiplies two vectors into a single number: a · b = a₁b₁ + a₂b₂. The recipe is easy. What the number means is the interesting part.
It measures how far one vector reaches along the other. Drop a perpendicular from the tip of b onto the line through a: the shadow is the projection of b on a. The dot product is the length of that shadow times the length of a.
So the sign tells you about direction. Positive: the vectors point broadly the same way. Zero: they are perpendicular. Negative: they point broadly against each other.
The same number can be written with the angle between the two: a · b = |a| |b| cos θ. For perpendicular vectors cos θ = 0, which is why the dot product vanishes.
Work in physics is a dot product: force times the distance moved in the direction of the force. When you pull a sled at an angle, only the part of your pull along the ground does work.
Answer it first
Are (2, −1) and (1, 2) perpendicular?
Show the answer
Yes: 2·1 + (−1)·2 = 0.
Reading is not learning it
Leo teaches this at your pace, asks for your attempt before he shows his, and checks a week later what stayed.