The Cross Product

Chapter 4 • Section 4-3

"When two vectors meet... a third one rises! ⚔️"

✖️ Definition

The cross product of $\vec{u} = (x_1, y_1, z_1)$ and $\vec{v} = (x_2, y_2, z_2)$ is a vector:

$$ \vec{u} \times \vec{v} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \end{vmatrix} $$

Result: $\vec{u} \times \vec{v}$ is orthogonal to both $\vec{u}$ and $\vec{v}$.

$\vec{u} \times \vec{v} = -(\vec{v} \times \vec{u})$ (Anti-commutative)
$\vec{u} \times (\vec{v} + \vec{w}) = \vec{u} \times \vec{v} + \vec{u} \times \vec{w}$ (Distributive)
$\vec{u} \times \vec{u} = \vec{0}$

$$ \norm{\vec{u} \times \vec{v}}^2 = \norm{\vec{u}}^2 \norm{\vec{v}}^2 - (\vec{u} \cdot \vec{v})^2 $$

The length equals the area of the parallelogram formed by $\vec{u}$ and $\vec{v}$:

$$ \norm{\vec{u} \times \vec{v}} = \norm{\vec{u}} \norm{\vec{v}} \sin \theta $$

Area of triangle with vertices $P, Q, R$:

$$ \text{Area} = \frac{1}{2} \norm{\vec{PQ} \times \vec{PR}} $$

Volume of parallelepiped formed by $\vec{u}, \vec{v}, \vec{w}$:

$$ V = |\vec{u} \cdot (\vec{v} \times \vec{w})| $$

Vector $\vec{u}$

Vector $\vec{v}$

Compute $\vec{i} \times \vec{j}$.