The dot product

The dot product takes two vectors and returns a plain number — not a vector. That is the first thing to fix in your head, because the name "product" suggests otherwise and the cross product, met soon after, really does return a vector. Written in coordinates it is just aₓbₓ + a_yb_y + a_zb_z; written geometrically it is |a||b|cos θ, where θ is the angle between them.

Those two forms together are what make it useful. The geometric form says the dot product measures alignment: largest when the vectors point the same way, zero when they are perpendicular, and negative when the angle is obtuse. The coordinate form lets you compute it without ever finding the angle. Set them equal and you can solve for θ — which is the standard way to find the angle between two vectors in space.

The perpendicularity test is where it earns its keep. Two non-zero vectors are at right angles precisely when their dot product is zero, and checking that is three multiplications and two additions with no trigonometry in sight. It is also the reason work in physics is a dot product: pushing at right angles to the motion does no work, and the cos θ factor is exactly what encodes that.