Cross Product (Vector Product) of Vectors and Properties – Example

 

The cross product, also known as the vector product, is an operation between two vectors that results in a third vector orthogonal to both of the original vectors. It is denoted by the symbol $\times$. Given two vectors $\mathbf{A}$ and $\mathbf{B}$, their cross product $\mathbf{C}$ can be calculated as:

\begin{equation}
\mathbf{C} = \mathbf{A} \times \mathbf{B}
\end{equation}

The cross product can be defined using the determinant of a 3×3 matrix:

\begin{equation}
\mathbf{C} = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
A_x & A_y & A_z \\
B_x & B_y & B_z
\end{vmatrix}
\end{equation}

where $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ are the unit vectors in the $x$, $y$, and $z$ directions, respectively. The components of the resulting vector $\mathbf{C}$ are given by:

\begin{align}
C_x &= A_yB_z – A_zB_y \\
C_y &= A_zB_x – A_xB_z \\
C_z &= A_xB_y – A_yB_x
\end{align}

 

Example

Let’s consider the vectors $\mathbf{A} = 3\mathbf{i} + 2\mathbf{j} – \mathbf{k}$ and $\mathbf{B} = 2\mathbf{i} + 4\mathbf{j} + \mathbf{k}$. We can calculate their cross product as follows:

\begin{align*}
C_x &= (2)(-1) – (4)(-1) = -2 + 4 = 2 \\
C_y &= (1)(3) – (2)(-1) = 3 + 2 = 5 \\
C_z &= (3)(4) – (2)(2) = 12 – 4 = 8
\end{align*}

Therefore, the cross product of $\mathbf{A}$ and $\mathbf{B}$ is $\mathbf{C} = 2\mathbf{i} + 5\mathbf{j} + 8\mathbf{k}$.

 

Properties

The cross product has several important properties:

1. The cross product of two vectors is orthogonal to both of the original vectors.
2. The magnitude of the cross product is equal to the area of the parallelogram formed by the original vectors.
3. The direction of the cross product follows the right-hand rule: if you curl the fingers of your right hand from the direction of $\mathbf{A}$ to $\mathbf{B}$, then your thumb points in the direction of $\mathbf{C}$.
4. The cross product is anti-commutative, meaning that $\mathbf{A} \times \mathbf{B} = -(\mathbf{B} \times \mathbf{A})$.
5. If the cross product of two vectors is zero, then the vectors are parallel or anti-parallel.