Geometric significance of the dot product

Parametric equations

The dot product, also called the inner product, is

The sign is the sign of $\cos \theta$: positive when the angle is acute, negative when it is obtuse. If either vector is the zero vector, there is no angle. If both are nonzero and the angle is a right angle, the product is zero.

The scalar projection of $\mathbf{u}$ onto $\mathbf{v}$, for $\mathbf{v}\ne 0$, is

The gold arrow is the vector projection of $\mathbf{u}$ onto the line of $\mathbf{v}$. Its signed length along $\mathbf{v}$ is that number. The same product is the work $W=\mathbf{F}\cdot \mathbf{d}$ of a force along a displacement. Switching to that reading only renames the two arrows.

Interaction

• Dot product u · v: drag either vector and watch the sign and size of $\mathbf{u}\cdot \mathbf{v}$ change with the angle
• Angle θ: mark $\theta$. When the vectors are perpendicular the product is zero and the badge is highlighted
• Projection: drop $\mathbf{u}$ onto the direction of $\mathbf{v}$ and compare that signed length with the dot product
• Work W = F · d: rename the arrows as a force and a displacement. The formula is the same

Related

• Vector addition and scalar multiplication
• Vector projection
• Basic patterns of a vector field

Further reading

• Dot product (Wikipedia)