If $\mathbf{u}$ and $\mathbf{v}$ are nonzero vectors in $R^{2}$ or $R^{3}$, and if $\theta$ is the angle between $\mathrm{u}$ and $\mathbf{v}$, then the dot product (also called the Euclidean inner product) of $\mathbf{u}$ and $\mathbf{v}$ is denoted by $\mathbf{u} \cdot \mathbf{v}$ and is defined as $$ \mathbf{u} \cdot \mathbf{v}=\|\mathbf{u}\|\|\mathbf{v}\| \cos \theta $$ If $\mathbf{u}=\mathbf{0}$ or $\mathbf{v}=\mathbf{0}$, then we define $\mathbf{u} \cdot \mathbf{v}$ to be 0 .