What does the dot product of two vectors actually mean?

The dot product a·b = |a||b|cos θ, where θ is the angle between the vectors. Geometrically it measures how much one vector points along the other: it is the length of a multiplied by the projection of b onto a. When the vectors are perpendicular, θ = 90°, cos θ = 0, so the dot product is zero, which is the standard test for perpendicularity. When the angle is acute the result is positive; when obtuse it is negative. In components, a·b = a₁b₁ + a₂b₂ + a₃b₃. A common use is finding the angle: cos θ = (a·b)/(|a||b|). In physics, work done equals force dotted with displacement, which is exactly this projection idea.