Dot Product

Definition: Vectors

Let

• $\mathbf{u},\mathbf{v}$ : Nonzero vectors in $R^2$ or $R^3$
• $\theta$ : Angle between $\mathbf{u}$ and $\mathbf{v}$

Then the dot product of $\mathbf{u}$ and $\mathbf{v}$ (denoted by $\mathbf{u}\cdot \mathbf{v}$) is defined as $\mathbf{u}\cdot \mathbf{v}=||\mathbf{u}||||\mathbf{v}||\cos \theta$

If $\mathbf{u}=0$ or $\mathbf{v}=0$ then we define $\mathbf{u}\cdot \mathbf{v}$ to be $0$

Remark

We can obtain the angle between $\mathbf{u}$ and $\mathbf{v}$ using:

$$\cos \theta =\frac{\mathbf{u}\cdot \mathbf{v}}{||\mathbf{u}||||\mathbf{v}||}$$