Cross product
Definition
The cross product is an operation between two vectors in three dimensions. If \(\vec u\) and \(\vec v\) are vectors in \(\mathbb{R}^3\), then \(\vec u \times \vec v\) is another vector in \(\mathbb{R}^3\).
Geometrically, the resulting vector is perpendicular to both original vectors. Its direction is determined by the right-hand rule.
$$\vec u=(u_1,u_2,u_3),\qquad \vec v=(v_1,v_2,v_3)$$
Formula by cofactors
A practical way to compute the cross product is to expand the following determinant by cofactors along the first row:
$$ \vec u\times\vec v= \begin{vmatrix} \hat \imath & \hat \jmath & \hat k\\ u_1 & u_2 & u_3\\ v_1 & v_2 & v_3 \end{vmatrix} $$
After expanding the determinant, we obtain:
$$ \vec u\times\vec v= (u_2v_3-u_3v_2)\hat \imath -(u_1v_3-u_3v_1)\hat \jmath +(u_1v_2-u_2v_1)\hat k $$
$$ \vec u\times\vec v= (u_2v_3-u_3v_2,\;u_3v_1-u_1v_3,\;u_1v_2-u_2v_1) $$
Example
Let \(\vec u=(2,-1,3)\) and \(\vec v=(1,4,-2)\). Then:
$$ \vec u\times\vec v= \begin{vmatrix} \hat \imath & \hat \jmath & \hat k\\ 2 & -1 & 3\\ 1 & 4 & -2 \end{vmatrix} $$
$$ \vec u\times\vec v =((-1)(-2)-3(4),\;3(1)-2(-2),\;2(4)-(-1)(1)) $$
$$ \vec u\times\vec v=(-10,7,9) $$
Important properties
- It is not commutative: \(\vec u\times\vec v=-(\vec v\times\vec u)\).
- A vector crossed with itself gives the zero vector: \(\vec u\times\vec u=\vec 0\).
- Its magnitude is \(\|\vec u\times\vec v\|=\|\vec u\|\,\|\vec v\|\sin\theta\).
- The magnitude represents the area of the parallelogram formed by the two vectors.
Practice activity
Use the following practice activity to generate random vectors and reveal the cofactor expansion step by step.
Practice: Cross product by cofactors
Generate a random exercise and reveal the solution one step at a time.
Instructions:
🧮 A random cross product exercise is generated.
👆 Press Show next step to reveal the cofactor solution gradually.
🎲 Press New exercise to generate different vectors.