Cross Product

Finding perpendicular vectors in three dimensions

The Perpendicular Problem

Given two vectors in 3D space, can we find a third vector that is perpendicular to both of them? This is exactly what the cross product computes. Unlike the dot product, which gives us a scalar, the cross product gives us a new vector.

Think of it this way: two vectors define a plane in 3D space. The cross product gives us a vector that points straight out of that plane—perpendicular to both input vectors simultaneously.

Interactive: Cross Product in 3D

Vector a

2.0
0.0
0.0

Vector b

0.0
2.0
0.0
a⃗=(2.0,0.0,0.0)\vec{a} = (2.0, 0.0, 0.0)b⃗=(0.0,2.0,0.0)\vec{b} = (0.0, 2.0, 0.0)a⃗×b⃗=(0.0,0.0,4.0)\vec{a} \times \vec{b} = (0.0, 0.0, 4.0)

The purple vector is perpendicular to both blue and red vectors. Rotate the view to verify.

The notation for the cross product uses a multiplication symbol: a⃗×b⃗\vec{a} \times \vec{b}. The result is always a vector, and it is always perpendicular to both a⃗\vec{a} and b⃗\vec{b}.

The Right-Hand Rule

But wait—if the cross product is perpendicular to both input vectors, there are actually two possible directions: one pointing up from the plane, and one pointing down. Which one do we get?

This is where the right-hand rule comes in. It tells us exactly which direction the cross product will point. The rule is simple: point your fingers along the first vector, curl them toward the second vector, and your thumb shows the direction of the result.

Interactive: Right-Hand Rule

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The Right-Hand Rule:

  1. Point your fingers in the direction of the first vector (blue, a⃗\vec{a})
  2. Curl them toward the second vector (red, b⃗\vec{b})
  3. Your thumb points in the direction of the cross product (purple)

Watch how the purple result vector always stays perpendicular to the plane containing a and b.

This convention is not arbitrary—it is deeply connected to the orientation of our coordinate system. In a right-handed coordinate system, the x-axis crossed with the y-axis gives the z-axis: x^×y^=z^\hat{x} \times \hat{y} = \hat{z}.

The Formula

To compute the cross product of two vectors a⃗=(a1,a2,a3)\vec{a} = (a_1, a_2, a_3) and b⃗=(b1,b2,b3)\vec{b} = (b_1, b_2, b_3), we use:

a⃗×b⃗=[a2b3−a3b2a3b1−a1b3a1b2−a2b1]\vec{a} \times \vec{b} = \begin{bmatrix} a_2 b_3 - a_3 b_2 \\ a_3 b_1 - a_1 b_3 \\ a_1 b_2 - a_2 b_1 \end{bmatrix}

The formula may look intimidating, but there is a pattern. Each component of the result is computed from the other two components of the inputs, cycling through in a specific order. A helpful mnemonic: for each component of the result, you skip that same component in the inputs and compute a 2×2 determinant from the remaining values.

Another way to remember: the cross product can be written as the determinant of a matrix with the unit vectors in the first row:

a⃗×b⃗=∣i^j^k^a1a2a3b1b2b3∣\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}

Magnitude as Area

The magnitude of the cross product has a beautiful geometric interpretation: it equals the area of the parallelogram formed by the two input vectors.

∣a⃗×b⃗∣=∣a⃗∣⋅∣b⃗∣⋅sin⁡(θ)|\vec{a} \times \vec{b}| = |\vec{a}| \cdot |\vec{b}| \cdot \sin(\theta)

where θ\theta is the angle between the vectors. When the vectors are parallel (θ=0\theta = 0 or θ=180°\theta = 180°), the cross product is zero because the parallelogram has no area.

Interactive: Parallelogram Area

Vector a

2.0
0.0
0.0

Vector b

1.0
2.0
0.0

Parallelogram Area

∣a⃗×b⃗∣=4.00|\vec{a} \times \vec{b}| = 4.00

The green parallelogram is formed by the two vectors. Its area equals the magnitude of the cross product.

This property makes the cross product incredibly useful in physics and graphics. Need to compute the area of a triangle? Take half the cross product magnitude. Need to find the torque from a force? Cross the position vector with the force vector.

Anticommutativity

Unlike multiplication of regular numbers, the order of vectors in a cross product matters. In fact, swapping the order reverses the direction:

a⃗×b⃗=−(b⃗×a⃗)\vec{a} \times \vec{b} = -(\vec{b} \times \vec{a})

This property is called anticommutativity. The right-hand rule explains why: when you swap which vector your fingers point along and which they curl toward, your thumb ends up pointing the opposite direction.

Interactive: Anticommutativity

Vector a

2.0
1.0
0.0

Vector b

0.0
2.0
1.0
a⃗×b⃗=(1.0,−2.0,4.0)\vec{a} \times \vec{b} = (1.0, -2.0, 4.0)b⃗×a⃗=(−1.0,2.0,−4.0)\vec{b} \times \vec{a} = (-1.0, 2.0, -4.0)

Notice how b × a points in exactly the opposite direction of a × b. Same magnitude, opposite direction.

A special case: crossing any vector with itself gives zero, because there is no unique perpendicular direction: a⃗×a⃗=0⃗\vec{a} \times \vec{a} = \vec{0}.

Applications

The cross product appears everywhere in physics, computer graphics, and engineering:

  • Surface normals: Given two edge vectors of a polygon, their cross product gives the normal vector for lighting calculations
  • Torque: The rotational force from an applied force is τ⃗=r⃗×F⃗\vec{\tau} = \vec{r} \times \vec{F}
  • Angular momentum: A particle's angular momentum is L⃗=r⃗×p⃗\vec{L} = \vec{r} \times \vec{p}
  • Magnetic force: The force on a moving charge is F⃗=qv⃗×B⃗\vec{F} = q\vec{v} \times \vec{B}

In each case, the cross product captures the relationship between two directions to produce a third direction that is perpendicular to both.

Connection to Determinants

The cross product formula looks remarkably like a determinant—and this is no coincidence. The 2D determinant measures signed area; the cross product magnitude measures area of a parallelogram in 3D. Both capture the idea of "how much space do these vectors span?"

The dot product and cross product complement each other: the dot product tells you how much two vectors align (related to cosine), while the cross product tells you how much they "span" (related to sine). Together they give you the full picture of the relationship between two vectors.

Key Takeaways

  • The cross product takes two 3D vectors and returns a vector perpendicular to both
  • The right-hand rule determines which of the two perpendicular directions the result points
  • The magnitude of the cross product equals the area of the parallelogram formed by the two input vectors
  • Order matters: a⃗×b⃗=−(b⃗×a⃗)\vec{a} \times \vec{b} = -(\vec{b} \times \vec{a})
  • The cross product only exists in 3D (and 7D, but that is another story)
  • The formula resembles a determinant, connecting to our study of area scaling