Vector (Cross) Product

Rotational quantities — torque, angular momentum, angular velocity — are all defined using the cross product of vectors. Unlike the dot product (which gives a scalar), the cross product gives another vector, perpendicular to both inputs. It encodes both magnitude and sense of rotation.

Concept

The vector (cross) product of two vectors $\vec{a}$ and $\vec{b}$ is

$\vec{a}\times\vec{b} = |\vec{a}||\vec{b}|\sin\theta\,\hat{n}$

where $\theta$ is the angle between them ($0 \le \theta \le \pi$) and $\hat{n}$ is a unit vector perpendicular to both, in the direction given by the right-hand rule: curl the fingers of the right hand from $\vec{a}$ to $\vec{b}$; the thumb points along $\hat{n}$.

Key properties:

• Anticommutative: $\vec{a}\times\vec{b} = -\vec{b}\times\vec{a}$.
• Distributive: $\vec{a}\times(\vec{b}+\vec{c}) = \vec{a}\times\vec{b} + \vec{a}\times\vec{c}$.
• $\vec{a}\times\vec{a} = \vec{0}$.
• $|\vec{a}\times\vec{b}|$ equals the area of the parallelogram with sides $\vec{a}$, $\vec{b}$.

In Cartesian components with unit vectors $\hat{i}$, $\hat{j}$, $\hat{k}$:

$\hat{i}\times\hat{j} = \hat{k}, \quad \hat{j}\times\hat{k} = \hat{i}, \quad \hat{k}\times\hat{i} = \hat{j}$

For $\vec{a} = a_x\hat{i}+a_y\hat{j}+a_z\hat{k}$ and similarly $\vec{b}$:

$\vec{a}\times\vec{b} = \begin{vmatrix}\hat{i} & \hat{j} & \hat{k}\\ a_x & a_y & a_z\\ b_x & b_y & b_z\end{vmatrix}$

$= (a_y b_z - a_z b_y)\hat{i} - (a_x b_z - a_z b_x)\hat{j} + (a_x b_y - a_y b_x)\hat{k}$

Use in rotation:

• Torque: $\vec{\tau} = \vec{r}\times\vec{F}$.
• Angular momentum: $\vec{L} = \vec{r}\times\vec{p}$.
• Velocity of rotating particle: $\vec{v} = \vec{\omega}\times\vec{r}$.

Derivation

Area of parallelogram interpretation. Two vectors $\vec{a}$ and $\vec{b}$ at angle $\theta$ form a parallelogram. Its base is $|\vec{a}|$ and height $|\vec{b}|\sin\theta$, so area $= |\vec{a}||\vec{b}|\sin\theta = |\vec{a}\times\vec{b}|$.

$\vec{v} = \vec{\omega}\times\vec{r}$. A particle at position $\vec{r}$ from the axis, with angular velocity $\vec{\omega}$ along the axis, moves with linear velocity perpendicular to both: magnitude $\omega r \sin\theta$ (where $\theta$ is angle between $\vec{\omega}$ and $\vec{r}$, and $r\sin\theta$ is the perpendicular distance from the axis). Direction is tangential, given by the right-hand rule.

Worked Example

Find $\vec{a}\times\vec{b}$ for $\vec{a} = 2\hat{i} + 3\hat{j} - \hat{k}$, $\vec{b} = \hat{i} - \hat{j} + 2\hat{k}$.

Solution:

$\vec{a}\times\vec{b} = \begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\ 2&3&-1\\ 1&-1&2\end{vmatrix}$

$= \hat{i}(3 \cdot 2 - (-1)(-1)) - \hat{j}(2 \cdot 2 - (-1)(1)) + \hat{k}(2(-1) - 3(1))$

$= \hat{i}(6 - 1) - \hat{j}(4 + 1) + \hat{k}(-2 - 3) = 5\hat{i} - 5\hat{j} - 5\hat{k}$

Check perpendicularity: $(5)(2) + (-5)(3) + (-5)(-1) = 10 - 15 + 5 = 0$. ✓ ($\vec{a}\times\vec{b}$ is perpendicular to $\vec{a}$.)

Common Confusions

• Cross product is not commutative. $\vec{a}\times\vec{b} = -\vec{b}\times\vec{a}$. Order matters.
• Sin, not cos. Dot product uses $\cos$; cross uses $\sin$. Parallel vectors have zero cross product; perpendicular vectors have maximum cross product magnitude.
• Right-hand rule is the convention for direction; left-handed would flip all signs.
• Magnitude not enough. A cross product needs both magnitude and direction; always determine the direction.

Key Takeaways

• $\vec{a}\times\vec{b} = |\vec{a}||\vec{b}|\sin\theta\,\hat{n}$; direction from right-hand rule.
• Anticommutative: $\vec{a}\times\vec{b} = -\vec{b}\times\vec{a}$.
• $|\vec{a}\times\vec{b}|$ = area of parallelogram with sides $\vec{a}$, $\vec{b}$.
• Components: determinant of the $3\times 3$ matrix.
• Underpins torque, angular momentum, and rotational velocity in physics.