PlanetPhysics/Vector Identities

It is difficult to get anywhere in physics without a firm understanding of vectors and their common operations. Here, we will give vector identities as a reference. Basic terminology to keep straight.

{\mathbf Vector Magnitude

$$ A = \left | \mathbf{A} \right | = \sqrt{{A_x}^2 + {A_y}^2 + {A_z}^2 }$$\\ $$ A = \sqrt{\mathbf{A} \cdot \mathbf{A}} $$

{\mathbf scalar product (Dot Product)}

$$ \mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y + A_z B_z$$ \\ $$ \mathbf{A} \cdot \mathbf{B} = \left | \mathbf{A} \right | \left | \mathbf{B} \right | \cos \theta$$

{\mathbf vector product (Cross Product)}

$$ \mathbf{A} \times \mathbf{B} = \left ( A_y B_z - A_z B_y \right ) \mathbf{\hat{i}} + \left ( A_z B_x - A_x B_z \right ) \mathbf{\hat{j}} + \left ( A_x B_y - A_y B_x \right ) \mathbf{\hat{k}}$$

It can be easier to remember with determinant formulation

\mathbf{A} \times \mathbf{B} = \left| \begin{matrix} \mathbf{\hat{i}} & \mathbf{\hat{j}} & \mathbf{\hat{k}} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{matrix}\right| = \left ( A_y B_z - A_z B_y \right ) \mathbf{\hat{i}} + \left ( A_z B_x - A_x B_z \right ) \mathbf{\hat{j}} + \left ( A_x B_y - A_y B_x \right ) \mathbf{\hat{k}}$$

{\mathbf Vector Triple Product, aka. BAC CAB}

{\mathbf scalar Triple Product}

{\mathbf Gradient}

{\mathbf Gradient identities}

\\ \\ \\ \\

{\mathbf Divergence}

$$ \nabla \cdot \mathbf{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} +\frac{\partial A_z}{\partial z}

{\mathbf Divergence of the cross product}

$$ \nabla \cdot \left ( \mathbf{A} \times \mathbf{B} \right ) = \mathbf{B} \cdot \left ( \nabla \times \mathbf{A} \right ) - \mathbf{A} \cdot \left ( \nabla \times \mathbf{B} \right ) $$

{\mathbf Divergence of the curl}

$$ \nabla \cdot \left ( \nabla \times \mathbf{A} \right ) = 0 $$

{\mathbf Laplacian Identities}

$$ \nabla \times \left ( \nabla \times \mathbf{A} \right ) = \nabla \left ( \nabla \cdot \mathbf{A} \right ) - \nabla^2 \mathbf{A} $$