What is the cross product of two vectors?

The cross product of two vectors u and v in three-dimensional space is a third vector, written u × v, that is perpendicular to both u and v. Its components are u2 v3 − u3 v2, u3 v1 − u1 v3, and u1 v2 − u2 v1. Its length equals the area of the parallelogram spanned by u and v, and its direction is given by the right-hand rule.

Why is the cross product only defined in three dimensions?

In three dimensions there is exactly one direction perpendicular to two independent vectors, up to sign, so a vector-valued product makes sense. In two dimensions the formula collapses to a single number, the signed area u1 v2 − u2 v1. In four or more dimensions the space perpendicular to two vectors has more than one dimension, so there is no single perpendicular direction to choose. A product with the cross product's properties exists only in dimensions three and seven.

How does the i, j, k determinant give the cross product?

Write the basis vectors i, j, k in the first row of a 3 × 3 array, the components of u in the second row, and the components of v in the third. Expanding along the first row as if it were a determinant gives (u2 v3 − u3 v2) i − (u1 v3 − u3 v1) j + (u1 v2 − u2 v1) k, and those three coefficients are the components of u × v. It is a mnemonic rather than a true determinant, because the top row contains vectors, but the expansion is mechanical and always produces the correct result.

What is the difference between the dot product and the cross product?

The dot product takes two vectors and returns a scalar that measures how aligned they are; it is zero when the vectors are perpendicular. The cross product takes two vectors in three-dimensional space and returns a vector perpendicular to both; it is zero when the vectors are parallel. The dot product is defined in any dimension, the cross product only in three.

Is the cross product commutative?

No. The cross product is anticommutative: v × u equals −(u × v), the vector of the same length pointing the opposite way. It is also not associative, so (u × v) × w and u × (v × w) are different in general. It is bilinear, meaning it distributes over addition and scalars can be pulled out of either factor.

Cross Product of Vectors Visualizer

Key Terms

Cross product — the vector

\mathbf{u} \times \mathbf{v}

built from two vectors in

\mathbb{R}^3

. It is perpendicular to both, and its length is the area of the parallelogram the two vectors span.

Component formula —

w_1 = u_2 v_3 - u_3 v_2

,

w_2 = u_3 v_1 - u_1 v_3

,

w_3 = u_1 v_2 - u_2 v_1

. Each component skips its own row and cross-multiplies the other two.

Determinant mnemonic — the symbolic

3 \times 3

determinant with

\mathbf{i}, \mathbf{j}, \mathbf{k}

in the first row,

\mathbf{u}

in the second and

\mathbf{v}

in the third. Expanding along the first row reproduces the component formula.

Cyclic order — the index pattern

1 \to 2 \to 3 \to 1

that decides which product carries the plus sign in each component.

Minor and cofactor — the

2 \times 2

determinant left after striking a row and a column, and that determinant with its position sign

(+, -, +)

attached.

Right-hand rule — the convention that fixes which of the two perpendicular directions

\mathbf{u} \times \mathbf{v}

points along.

Anticommutativity —

\mathbf{v} \times \mathbf{u} = -(\mathbf{u} \times \mathbf{v})

. Swapping the operands reverses every component.

Getting Started with the Visualizer

Pick a method, then watch

\mathbf{u} \times \mathbf{v} = \mathbf{w}

fill one component at a time.

• Use the Method pills to switch between the component formula and the determinant expansion
• There is no length control: the cross product is defined only for vectors in

\mathbb{R}^3

, so

\mathbf{u}

,

\mathbf{v}

and

\mathbf{w}

always have three components
• Hover the ? icon for a reminder of what the cross product is and how the two methods relate
• Press play or step manually through the scene player; the speed selector and step log let you control pace and review
• Both methods produce the same three expressions — switch between them at the same step to compare

The Two Methods

The visualizer offers two routes to the same three numbers.

• Component formula — for component

k

, skip row

k

of both vectors and cross-multiply the remaining two rows:

w_k = u_a v_b - u_b v_a

, where

a

and

b

are the rows after

k

in cyclic order. Three scenes, one per component
• Determinant expansion — write

\mathbf{i}, \mathbf{j}, \mathbf{k}

above

\mathbf{u}

and

\mathbf{v}

as a symbolic determinant and expand along the first row. Each scene strikes a row and a column, highlights the

2 \times 2

minor that remains, applies the cofactor sign, and writes the result into

\mathbf{w}

The component formula is faster by hand. The determinant is easier to remember, and it explains where the minus sign in the middle component comes from: it is the cofactor sign of the second column.

The Opening Scene: Two Vectors in Three Dimensions

The player starts with

\mathbf{u}

and

\mathbf{v}

drawn as three-component columns and an empty

\mathbf{w}

waiting to hold

\mathbf{u} \times \mathbf{v}

.

Nothing is computed yet. What the scene fixes is the shape of the answer: two vectors in, a vector out, with the same three rows as its inputs.

Opening scene, frozen

u and v as three-component columns, w empty on the right. Two vectors in, one vector out - and the length is pinned at three because the operation exists nowhere else.

That output shape separates the cross product from the dot product, which takes the same two inputs and collapses them to a single number. Here the result keeps its three components, and each one will be assembled from entries of both

\mathbf{u}

and

\mathbf{v}

.

The one precondition is that both vectors live in

\mathbb{R}^3

. A cross product of two vectors in the plane, or in four dimensions, is not defined, which is why the tool offers no length control.

The Component Sweep

Each step of the component method skips one row of both vectors and cross-multiplies the other two: the

\mathbf{u}

entry from the first remaining row with the

\mathbf{v}

entry from the second, minus the reverse pairing.

The frozen picture below is the middle step. Row 2 is skipped, the entries in rows 3 and 1 are active, and

w_2 = u_3 v_1 - u_1 v_3

is being written.

Component method, middle step

Row 2 of u and v muted, rows 3 and 1 active, and w₂ being written as u₃v₁ − u₁v₃. The rows are read in cyclic order, 3 then 1, which is where the sign is usually lost.

Each component is a

2 \times 2

determinant. Take the two rows of

\mathbf{u}

and

\mathbf{v}

that are not the component's own row, arrange them as a

2 \times 2

block, and the component is that block's determinant, with the rows read in cyclic order.

The cyclic order is the whole difficulty. For

w_1

the rows run

2, 3

; for

w_3

they run

1, 2

; for

w_2

they run

3, 1

, not

1, 3

. Reading them as

1, 3

produces

u_1 v_3 - u_3 v_1

, the correct value with the wrong sign, and it is by far the most common cross product error.

The Determinant Expansion

The determinant method writes

\mathbf{i}, \mathbf{j}, \mathbf{k}

in the first row,

\mathbf{u}

in the second and

\mathbf{v}

in the third, then expands along the first row. Each step strikes the first row and one column, leaving a

2 \times 2

minor, and the cofactor sign for that column decides whether the minor is taken as is or negated.

The frozen picture below is the

\mathbf{j}

step. The first row and second column are struck, the minor is the block

u_1, u_3

over

v_1, v_3

, and the sign is minus.

Determinant method, the j step

First row and second column struck through, the 2×2 minor highlighted, and the cofactor sign for this column negative. The negated minor reorders into the same expression the component method produces.

This is a mnemonic rather than a determinant in the strict sense, because the top row holds vectors, not numbers. But the expansion is mechanical and it produces exactly the three expressions of the component formula, so anyone who can expand a

3 \times 3

determinant can compute a cross product without memorising the cyclic rule.

It also explains the awkward sign. The cofactor signs along the first row alternate

+, -, +

, so the

\mathbf{j}

coefficient is

-(u_1 v_3 - u_3 v_1)

. Distribute the minus and the terms reorder to

u_3 v_1 - u_1 v_3

, which is precisely the cyclic form. Two methods, one answer.

The Completed Product

The final scene fills all three rows, so

\mathbf{w}

reads

(u_2 v_3 - u_3 v_2,\; u_3 v_1 - u_1 v_3,\; u_1 v_2 - u_2 v_1)

.

Each entry mixes components of both inputs, and no entry of

\mathbf{w}

uses the matching entries of

\mathbf{u}

and

\mathbf{v}

: row 1 of

\mathbf{w}

never sees

u_1

or

v_1

.

Completed cross product, frozen

All three rows of w filled. Each holds a 2×2 determinant from the other two rows of u and v; the vector is perpendicular to both inputs and its length is the area they span.

Three facts follow from the completed picture. The vector is perpendicular to both inputs: dot it with

\mathbf{u}

or with

\mathbf{v}

and every product cancels against another, leaving zero. Its length is the area of the parallelogram spanned by

\mathbf{u}

and

\mathbf{v}

, which is

\|\mathbf{u}\|\,\|\mathbf{v}\| \sin\theta

. And its direction follows the right-hand rule, so swapping the inputs reverses every sign:

\mathbf{v} \times \mathbf{u} = -(\mathbf{u} \times \mathbf{v})

.

The perpendicularity is the property worth remembering as a check. A cross product computed by hand should always be dotted back against its two inputs, and both dot products must vanish.

Reading the Scene Player

Each scene combines highlights, arrows, and a caption.

• In the component method, the two active entries of

\mathbf{u}

are highlighted primary, the two active entries of

\mathbf{v}

secondary, and the skipped row of each vector is muted; four curved arrows flow into the destination component of

\mathbf{w}

, highlighted accent
• In the determinant method, the pivot basis vector is highlighted primary, the struck row and column are greyed and lined through, the

2 \times 2

minor is secondary, and one arrow flows from the pivot into the destination component
• Filled components of

\mathbf{w}

show their symbolic content,

u_a v_b - u_b v_a

, in a wider cell so the two-term expression stays readable
• The step log on the right keeps a record of every completed component, and clicking any entry jumps back to that scene

Why the Length Is Fixed at Three

Every other vector tool in this section has a length stepper. This one does not, and the reason is mathematical rather than a design choice.

• In

\mathbb{R}^3

there is exactly one direction perpendicular to two independent vectors (up to sign), so a vector-valued product makes sense
• In

\mathbb{R}^2

there is no room: the only thing left of the formula is the single number

u_1 v_2 - u_2 v_1

, the signed area, which is a scalar
• In

\mathbb{R}^n

for

n \geq 4

the space perpendicular to two vectors has dimension

n - 2

, so there is no single perpendicular direction to pick
• The generalization that does work in every dimension is the wedge product, which returns an oriented area rather than a vector; the cross product is the special case where that area can be encoded as a normal vector

A bilinear, anticommutative, vector-valued product of two vectors with the cross product's properties exists only in dimensions three and seven.

What the Cross Product Is

The cross product of two vectors

\mathbf{u}, \mathbf{v} \in \mathbb{R}^3

is the vector

\mathbf{u} \times \mathbf{v} = \begin{pmatrix} u_2 v_3 - u_3 v_2 \\ u_3 v_1 - u_1 v_3 \\ u_1 v_2 - u_2 v_1 \end{pmatrix}

The same vector is the formal expansion of

\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}

Geometrically,

\mathbf{u} \times \mathbf{v}

is perpendicular to both

\mathbf{u}

and

\mathbf{v}

, its length is

\|\mathbf{u} \times \mathbf{v}\| = \|\mathbf{u}\|\,\|\mathbf{v}\| \sin\theta

the area of the parallelogram spanned by the two vectors, and its direction is given by the right-hand rule: curl the fingers of the right hand from

\mathbf{u}

toward

\mathbf{v}

and the thumb points along

\mathbf{u} \times \mathbf{v}

.

For the full treatment, including the geometric proof of the length formula and the scalar triple product, see the cross product theory page.

Key Properties

The cross product is bilinear but neither commutative nor associative.

• Anticommutativity:

\mathbf{v} \times \mathbf{u} = -(\mathbf{u} \times \mathbf{v})

• Distributivity:

\mathbf{u} \times (\mathbf{v} + \mathbf{w}) = \mathbf{u} \times \mathbf{v} + \mathbf{u} \times \mathbf{w}

• Scalar pull-out:

(k\mathbf{u}) \times \mathbf{v} = k(\mathbf{u} \times \mathbf{v}) = \mathbf{u} \times (k\mathbf{v})

• Self-product:

\mathbf{u} \times \mathbf{u} = \mathbf{0}

, and more generally

\mathbf{u} \times \mathbf{v} = \mathbf{0}

exactly when

\mathbf{u}

and

\mathbf{v}

are parallel
• Perpendicularity:

\mathbf{u} \cdot (\mathbf{u} \times \mathbf{v}) = 0

and

\mathbf{v} \cdot (\mathbf{u} \times \mathbf{v}) = 0

• Not associative:

(\mathbf{u} \times \mathbf{v}) \times \mathbf{w} \neq \mathbf{u} \times (\mathbf{v} \times \mathbf{w})

in general
• Lagrange identity:

\|\mathbf{u} \times \mathbf{v}\|^2 = \|\mathbf{u}\|^2 \|\mathbf{v}\|^2 - (\mathbf{u} \cdot \mathbf{v})^2

• Standard basis:

\mathbf{i} \times \mathbf{j} = \mathbf{k}

,

\mathbf{j} \times \mathbf{k} = \mathbf{i}

,

\mathbf{k} \times \mathbf{i} = \mathbf{j}

, cyclically

The basis products are the component formula in miniature: the cyclic order

1 \to 2 \to 3 \to 1

is the same pattern that assigns the plus sign in each component.

Why It Matters

The cross product is the standard way to manufacture a perpendicular direction from two known ones.

• Normal vectors: the normal to the plane through three points, or to a surface patch, is a cross product of two edge vectors
• Area:

\|\mathbf{u} \times \mathbf{v}\|

is the area of the parallelogram spanned by

\mathbf{u}

and

\mathbf{v}

, and half of it is the area of the triangle
• Volume: the scalar triple product

\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})

is the signed volume of the parallelepiped, and it is the

3 \times 3

determinant with the three vectors as rows
• Physics: torque

\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}

, angular momentum

\mathbf{L} = \mathbf{r} \times \mathbf{p}

, and the magnetic force

q\mathbf{v} \times \mathbf{B}

are all cross products
• Orientation: the sign of a cross product tells whether a turn is clockwise or counterclockwise, which drives orientation tests in computational geometry
• Computer graphics: lighting, back-face culling and camera coordinate frames all rely on cross products for normals and perpendicular axes

Worked Example

Take

\mathbf{u} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}, \quad \mathbf{v} = \begin{pmatrix} 4 \\ 5 \\ 6 \end{pmatrix}

Component by component:

w_1 = u_2 v_3 - u_3 v_2 = 2 \cdot 6 - 3 \cdot 5 = 12 - 15 = -3

w_2 = u_3 v_1 - u_1 v_3 = 3 \cdot 4 - 1 \cdot 6 = 12 - 6 = 6

w_3 = u_1 v_2 - u_2 v_1 = 1 \cdot 5 - 2 \cdot 4 = 5 - 8 = -3

So

\mathbf{u} \times \mathbf{v} = \begin{pmatrix} -3 \\ 6 \\ -3 \end{pmatrix}

Check perpendicularity with the dot product:

\mathbf{u} \cdot \mathbf{w} = -3 + 12 - 9 = 0

and

\mathbf{v} \cdot \mathbf{w} = -12 + 30 - 18 = 0

. The area of the parallelogram spanned by

\mathbf{u}

and

\mathbf{v}

is

\|\mathbf{w}\| = \sqrt{9 + 36 + 9} = \sqrt{54} = 3\sqrt{6}

.

Step through the visualizer with either method to see the same three expressions assembled symbolically.

Common Mistakes

A few mistakes recur.

• Losing the sign of the middle component —

w_2 = u_3 v_1 - u_1 v_3

, not

u_1 v_3 - u_3 v_1

. The cyclic rule puts row 3 before row 1; the determinant explains it as the minus cofactor sign of the second column
• Treating it as commutative —

\mathbf{v} \times \mathbf{u}

is

-(\mathbf{u} \times \mathbf{v})

, the opposite vector
• Confusing it with the dot product — the dot product returns a scalar and measures alignment; the cross product returns a vector and measures perpendicularity and area
• Assuming associativity —

(\mathbf{u} \times \mathbf{v}) \times \mathbf{w}

and

\mathbf{u} \times (\mathbf{v} \times \mathbf{w})

are different vectors in general
• Applying it outside $\mathbb{R}^3$ — there is no vector cross product of two vectors in

\mathbb{R}^2

or

\mathbb{R}^4

; in the plane the formula collapses to the scalar

u_1 v_2 - u_2 v_1

• Reading the determinant literally — the top row holds vectors, so it is a mnemonic for the expansion, not a determinant of a numerical matrix
• Skipping the check — a cross product is easy to verify: dot the result with each input, and both must come out zero

Related Concepts

Dot product — the other product of two vectors; scalar-valued, and zero exactly when the vectors are perpendicular, which makes it the natural check on a cross product.

Determinant — the

2 \times 2

determinants inside each component, and the

3 \times 3

mnemonic that organises them.

Scalar triple product —

\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})

, the signed volume of a parallelepiped.

Normal vector — a vector perpendicular to a plane or surface; cross products are the usual way to produce one.

Right-hand rule — the orientation convention that picks the sign of the cross product.

Orthogonality — the relationship the cross product guarantees between its output and each of its inputs.

Wedge product — the generalization to any dimension, returning an oriented area rather than a vector.

Levi-Civita symbol — the index notation

w_i = \varepsilon_{ijk} u_j v_k

that encodes the cyclic sign rule.

Magnitude —

\|\mathbf{u} \times \mathbf{v}\|

is the area of the parallelogram the two vectors span, so the length of the result carries geometric information the components alone do not.

Span and independence — two vectors in

\mathbb{R}^3

are linearly dependent exactly when their cross product is the zero vector, which makes it the

\mathbb{R}^3

analogue of the determinant test in the plane.

Cross Product of Vectors

Step explanations