# Definitions Let $\mathbb{R}^{3}$ denote the [[Real n-Space|real 3-space]] and $\underline{i},\underline{j},\underline{k}$ denote its [[Standard basis of real n-space|standard basis]]. Let $\underline{u}=u_{1}\underline{i}+u_{2}\underline{j}+u_{3}\underline{k}, \underline{v}=v_{1}\underline{i} + v_{2}\underline{j} + v_{3}\underline{k} \in \mathbb{R}^{3}.$ > [!NOTE] Definition 1 > The vector cross product, denoted $\underline{u}\times \underline{v},$ is defined as $\underline{u} \times \underline{v} = \begin{vmatrix}\underline{i} & \underline{j} & \mathbf k\\ u_{1} & u_{2} & u_{3} \\ v_{1} & v_{2} & v_{3} \\ \end{vmatrix}$where $|\dots|$ denotes [[Determinant|determinant]]. > [!NOTE] Definiton 2 > The vector cross product, denoted $\underline{u}\times \underline{v},$ is defined as $\underline{u} \times \underline{v} = ||\underline{u}|| \; ||\underline{v}|| \sin \theta \; \hat{\underline{n}}$where $||\underline{u}||$ denotes the [[Euclidean Norm|length]] of $\underline{u}$; $\theta$ denotes the [[Angle Between Nonzero Real Vectors|angle]] $\angle\underline{uv}$; and $\hat{\underline{n}}$ denotes the [[Unit Vector in Direction of Non-Zero Real Vector|unit vector]] that is [[Orthogonal Subset of Real n-Space|orthogonal]] to both $\underline{u}$ and $\underline{v}$ in the direction of the [[Right-Hand Rule for Cross Product in Real 3-Space|right hand rule]]. **Note**: By # Properties By [[Cross Product in Real 3-Space is Anti-commutative]], for all $\underline{u},\underline{v} \in \mathbb{R}^{3},$ $\underline{u}\times \underline{v}=-\underline{v}\times \underline{u}.$