Cross Product (Vector Product)
The cross product turns two vectors into a third vector perpendicular to both. It encodes rotational quantities — torque, angular momentum, magnetic force.
Concept
Definition
For vectors A ⃗ \vec{A} A B ⃗ \vec{B} B θ \theta θ A ⃗ × B ⃗ = A B sin θ n ^ , \vec{A}\times\vec{B} = AB\sin\theta\,\hat{n}, A × B = A B sin θ n ^ , n ^ \hat{n} n ^ A ⃗ \vec{A} A B ⃗ \vec{B} B right-hand rule .
Right-Hand Rule
Point your fingers along A ⃗ \vec{A} A B ⃗ \vec{B} B A ⃗ × B ⃗ \vec{A}\times\vec{B} A × B
A ⃗ × B ⃗ = ∣ i ^ j ^ k ^ A x A y A z B x B y B z ∣ \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 × B = i ^ A x B x j ^ A y B y k ^ A z B z
Expanding:
A ⃗ × B ⃗ = ( A y B z − A z B y ) i ^ − ( A x B z − A z B x ) j ^ + ( A x B y − A y B x ) k ^ . \vec{A}\times\vec{B} = (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}. A × B = ( A y B z − A z B y ) i ^ − ( A x B z − A z B x ) j ^ + ( A x B y − A y B x ) k ^ .
Identities of Unit Vectors
Cross Result i ^ × j ^ \hat{i}\times\hat{j} i ^ × j ^ k ^ \hat{k} k ^ j ^ × k ^ \hat{j}\times\hat{k} j ^ × k ^ i ^ \hat{i} i ^ k ^ × i ^ \hat{k}\times\hat{i} k ^ × i ^ j ^ \hat{j} j ^ j ^ × i ^ \hat{j}\times\hat{i} j ^ × i ^ − k ^ -\hat{k} − k ^ i ^ × i ^ \hat{i}\times\hat{i} i ^ × i ^ 0 ⃗ \vec{0} 0
The cyclic pattern: i → j → k → i i \to j \to k \to i i → j → k → i
Properties
Property Statement Anti-commutative A ⃗ × B ⃗ = − B ⃗ × A ⃗ \vec{A}\times\vec{B} = -\vec{B}\times\vec{A} A × B = − B × A Distributive A ⃗ × ( B ⃗ + C ⃗ ) = A ⃗ × B ⃗ + A ⃗ × C ⃗ \vec{A}\times(\vec{B}+\vec{C}) = \vec{A}\times\vec{B} + \vec{A}\times\vec{C} A × ( B + C ) = A × B + A × C Self-cross zero A ⃗ × A ⃗ = 0 ⃗ \vec{A}\times\vec{A} = \vec{0} A × A = 0 Parallel test A ⃗ × B ⃗ = 0 ⃗ \vec{A}\times\vec{B} = \vec{0} A × B = 0
Geometric Meaning
∣ A ⃗ × B ⃗ ∣ |\vec{A}\times\vec{B}| ∣ A × B ∣ A ⃗ \vec{A} A B ⃗ \vec{B} B
Applications
Torque: τ ⃗ = r ⃗ × F ⃗ \vec{\tau} = \vec{r}\times\vec{F} τ = r × F Angular momentum: L ⃗ = r ⃗ × p ⃗ \vec{L} = \vec{r}\times\vec{p} L = r × p Magnetic force on a charge: F ⃗ = q v ⃗ × B ⃗ \vec{F} = q\vec{v}\times\vec{B} F = q v × B Area vector for a planar loop.
Worked Example
Q1: A ⃗ = 2 i ^ + 3 j ^ \vec{A} = 2\hat{i} + 3\hat{j} A = 2 i ^ + 3 j ^ B ⃗ = i ^ − j ^ + 2 k ^ \vec{B} = \hat{i} - \hat{j} + 2\hat{k} B = i ^ − j ^ + 2 k ^ A ⃗ × B ⃗ \vec{A}\times\vec{B} A × B
Solution:
A ⃗ × B ⃗ = ∣ i ^ j ^ k ^ 2 3 0 1 − 1 2 ∣ \vec{A}\times\vec{B} = \begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\2&3&0\\1&-1&2\end{vmatrix} A × B = i ^ 2 1 j ^ 3 − 1 k ^ 0 2
i ^ \hat{i} i ^ 3 ( 2 ) − 0 ( − 1 ) = 6 3(2) - 0(-1) = 6 3 ( 2 ) − 0 ( − 1 ) = 6 j ^ \hat{j} j ^ − ( 2 ( 2 ) − 0 ( 1 ) ) = − 4 -(2(2) - 0(1)) = -4 − ( 2 ( 2 ) − 0 ( 1 )) = − 4 k ^ \hat{k} k ^ 2 ( − 1 ) − 3 ( 1 ) = − 5 2(-1) - 3(1) = -5 2 ( − 1 ) − 3 ( 1 ) = − 5
So A ⃗ × B ⃗ = 6 i ^ − 4 j ^ − 5 k ^ \vec{A}\times\vec{B} = 6\hat{i} - 4\hat{j} - 5\hat{k} A × B = 6 i ^ − 4 j ^ − 5 k ^
Q2: Torque from F ⃗ = 3 i ^ \vec{F} = 3\hat{i} F = 3 i ^ r ⃗ = 2 j ^ \vec{r} = 2\hat{j} r = 2 j ^ τ ⃗ = r ⃗ × F ⃗ = 2 j ^ × 3 i ^ = 6 ( j ^ × i ^ ) = − 6 k ^ N⋅m . \vec{\tau} = \vec{r}\times\vec{F} = 2\hat{j}\times 3\hat{i} = 6(\hat{j}\times\hat{i}) = -6\hat{k}\,\text{N·m}. τ = r × F = 2 j ^ × 3 i ^ = 6 ( j ^ × i ^ ) = − 6 k ^ N⋅m .
Common Confusions
"A ⃗ × B ⃗ = B ⃗ × A ⃗ \vec{A}\times\vec{B} = \vec{B}\times\vec{A} A × B = B × A
"A ⃗ × B ⃗ = 0 \vec{A}\times\vec{B} = 0 A × B = 0 A ⃗ = 0 ⃗ \vec{A} = \vec{0} A = 0 A ⃗ ∥ B ⃗ \vec{A}\parallel\vec{B} A ∥ B
"Cross product gives a scalar." — No, it gives a vector.
Wrong sign on j ^ \hat{j} j ^ + , − , + +, -, + + , − , +
Key Takeaways
A ⃗ × B ⃗ = A B sin θ n ^ \vec{A}\times\vec{B} = AB\sin\theta\,\hat{n} A × B = A B sin θ n ^ Magnitude equals the area of the parallelogram spanned by the two vectors.
Anti-commutative; distributive; self-cross is zero.
Computed via the 3 × 3 3\times 3 3 × 3
Cornerstone of rotational and magnetic physics.