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Cross Product — Definition, Formula & Examples

Cross Product

A way of multiplying two vectors, written u × v, in which the product is another vector. The cross product of two vectors results in a vector which is orthogonal to both the vectors being multiplied. The magnitude of the cross product of two vectors is found by the formula |u × v| = |u| |v| sin θ, where θ is the smaller angle between the vectors.

Note: Cross products are not commutative. That is, u × v ≠ v × u. The vectors u × v and v × u have the same magnitude but point in opposite directions.

 

Diagram showing vectors u and v with angle θ between them, and their cross product u×v pointing perpendicular (upward) to both.

 

 

See also

Dot product, triple product , inner product

Key Formula

u×v=∣ijku1u2u3v1v2v3∣=(u2v3−u3v2) i−(u1v3−u3v1) j+(u1v2−u2v1) k\mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix} = (u_2 v_3 - u_3 v_2)\,\mathbf{i} - (u_1 v_3 - u_3 v_1)\,\mathbf{j} + (u_1 v_2 - u_2 v_1)\,\mathbf{k}
Where:
  • u\mathbf{u} = First vector, with components (u₁, u₂, u₃)
  • v\mathbf{v} = Second vector, with components (v₁, v₂, v₃)
  • i,j,k\mathbf{i}, \mathbf{j}, \mathbf{k} = Unit vectors along the x, y, and z axes
  • ∣u×v∣=∣u∣∣v∣sin⁡θ|\mathbf{u} \times \mathbf{v}| = |\mathbf{u}||\mathbf{v}|\sin\theta = The magnitude formula, where θ is the angle between the two vectors

Worked Example

Problem: Find the cross product of u = (2, 3, 4) and v = (5, 6, 7).
Step 1: Set up the determinant with the unit vectors in the first row, u's components in the second row, and v's components in the third row.
u×v=∣ijk234567∣\mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 2 & 3 & 4 \\ 5 & 6 & 7 \end{vmatrix}
Step 2: Compute the i-component: multiply the middle entries diagonally and subtract.
i(3⋅7−4⋅6)=i(21−24)=−3 i\mathbf{i}(3 \cdot 7 - 4 \cdot 6) = \mathbf{i}(21 - 24) = -3\,\mathbf{i}
Step 3: Compute the j-component. Remember the j-term carries a negative sign in front.
−j(2⋅7−4⋅5)=−j(14−20)=6 j-\mathbf{j}(2 \cdot 7 - 4 \cdot 5) = -\mathbf{j}(14 - 20) = 6\,\mathbf{j}
Step 4: Compute the k-component using the first two entries of each vector row.
k(2⋅6−3⋅5)=k(12−15)=−3 k\mathbf{k}(2 \cdot 6 - 3 \cdot 5) = \mathbf{k}(12 - 15) = -3\,\mathbf{k}
Step 5: Combine all three components to write the final cross product vector.
u×v=(−3,  6,  −3)\mathbf{u} \times \mathbf{v} = (-3,\; 6,\; -3)
Answer: u × v = (−3, 6, −3)

Another Example

This example uses standard basis vectors to show a geometric application: the magnitude of the cross product gives the area of the parallelogram spanned by two vectors. It also illustrates the right-hand rule — crossing i into j yields k.

Problem: Find the area of the parallelogram formed by vectors a = (1, 0, 0) and b = (0, 1, 0).
Step 1: Compute the cross product a × b using the determinant formula.
a×b=∣ijk100010∣\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{vmatrix}
Step 2: Evaluate each component: i(0·0 − 0·1) − j(1·0 − 0·0) + k(1·1 − 0·0).
a×b=0 i−0 j+1 k=(0,0,1)\mathbf{a} \times \mathbf{b} = 0\,\mathbf{i} - 0\,\mathbf{j} + 1\,\mathbf{k} = (0, 0, 1)
Step 3: The area of the parallelogram equals the magnitude of the cross product.
∣a×b∣=02+02+12=1|\mathbf{a} \times \mathbf{b}| = \sqrt{0^2 + 0^2 + 1^2} = 1
Answer: The area of the parallelogram is 1 square unit. The cross product vector (0, 0, 1) = k points along the z-axis, confirming it is perpendicular to both a and b.

Frequently Asked Questions

What is the difference between the cross product and the dot product?
The cross product of two vectors produces a new vector that is perpendicular to both inputs, while the dot product produces a scalar (a single number). The cross product uses sine of the angle between the vectors, whereas the dot product uses cosine. Use the cross product when you need a perpendicular direction or an area; use the dot product when you need a projection or want to check if vectors are perpendicular.
Why is the cross product not commutative?
Swapping the order of the vectors reverses the direction of the result: u × v = −(v × u). This is called anticommutativity. The magnitudes are identical, but the two products point in opposite directions. The right-hand rule determines which direction the result points for a given order.
When is the cross product equal to zero?
The cross product is the zero vector when the two input vectors are parallel (or antiparallel), because sin 0° = 0 and sin 180° = 0. It is also zero if either vector is the zero vector. A zero cross product means the two vectors do not span a plane — they lie along the same line.

Cross Product vs. Dot Product

Cross ProductDot Product
Result typeA vectorA scalar (number)
Formula (magnitude)|u||v| sin θ|u||v| cos θ
CommutativityNot commutative (anticommutative)Commutative
Geometric meaningArea of the parallelogram; perpendicular directionProjection of one vector onto another
Zero whenVectors are parallelVectors are perpendicular
Dimensions required3D only (or 7D)Any number of dimensions

Why It Matters

You encounter the cross product in physics whenever torque, angular momentum, or the magnetic force on a charged particle is calculated — all of these are defined using cross products. In multivariable calculus, it appears when finding normal vectors to surfaces and computing surface integrals. Understanding the cross product is also essential in computer graphics, where it is used to determine surface normals for lighting calculations and to check the orientation of triangles.

Common Mistakes

Mistake: Forgetting the negative sign on the j-component in the determinant expansion.
Correction: The cofactor expansion alternates signs: +i, −j, +k. The j-component formula is −(u₁v₃ − u₃v₁), not +(u₁v₃ − u₃v₁). Double-check by expanding the 3×3 determinant carefully.
Mistake: Treating the cross product as commutative (assuming u × v = v × u).
Correction: The cross product is anticommutative: u × v = −(v × u). Switching the order flips the direction of the resulting vector. Always pay attention to the order of the vectors.

Related Terms