Dot Product

Lesson · Intermediate

Geometry: Coordinate Geometry

For vectors

\overrightarrow{u}=(u_1,u_2,\ldots,u_n),\qquad \overrightarrow{v}=(v_1,v_2,\ldots,v_n)

the dot product is

\overrightarrow{u}\cdot\overrightarrow{v}=u_1v_1+u_2v_2+\cdots+u_nv_n

The result is a number, not a vector.

If θ is the angle between the two vectors, then

\overrightarrow{u}\cdot\overrightarrow{v}=|\overrightarrow{u}||\overrightarrow{v}|\cos\theta

Therefore,

\cos\theta=\frac{\overrightarrow{u}\cdot\overrightarrow{v}}{|\overrightarrow{u}||\overrightarrow{v}|}

|\overrightarrow{u}+\overrightarrow{v}|^2=|\overrightarrow{u}|^2+|\overrightarrow{v}|^2+2\overrightarrow{u}\cdot\overrightarrow{v}

and

|\overrightarrow{u}-\overrightarrow{v}|^2=|\overrightarrow{u}|^2+|\overrightarrow{v}|^2-2\overrightarrow{u}\cdot\overrightarrow{v}

From these,

|\overrightarrow{u}+\overrightarrow{v}|^2+|\overrightarrow{u}-\overrightarrow{v}|^2=2|\overrightarrow{u}|^2+2|\overrightarrow{v}|^2

This last identity is the Parallelogram Law.