Cross Product: Building a Perpendicular from Two Vectors

The surface normal in 3D graphics, the torque on a motor, the orientation of a drone in flight - all computed via the cross product. Unity and Unreal Engine call it every frame to compute lighting for every triangle in the scene.

• 3D graphics: triangle normal for lighting = cross product of two edge vectors
• Robotics: torque τ = r × F controls joint angles in a robot arm
• Aviation: angular momentum and gyroscopic effects use the cross product
• Computer vision: orientation of a plane in 3D reconstruction from stereo cameras
• Physics: Lorentz force on a charge in a magnetic field - F = qv × B

Cross Product: Building a Perpendicular from Two Vectors

**Every surface in a 3D game knows which way it faces.** Without that knowledge, lighting is impossible: is light hitting the front or the back? At what angle? The GPU performs this calculation for billions of pixels per second, and at its core sits the **cross product**. The operation takes two vectors lying in a plane and produces a third vector perpendicular to both. The dot product gave a number - it measured an angle. The cross product gives a vector - it builds a direction.

The **cross product** is the answer to one question that keeps showing up: *give me a vector perpendicular to these two*. Unity asks it every frame for 3D lighting. A drone's IMU asks it 400× per second to track attitude. SLAM asks it for every camera pose estimate. One formula, three billion-dollar applications - and read as a rotation-generator (not just "right-hand rule"), they stop looking like three problems and start looking like one.

Intuition: perpendicular to a plane

Intuition: perpendicular to a plane

Two vectors a and b define a plane. The vector c = a x b is perpendicular to that plane: c is perpendicular to a and to b simultaneously. The length of c equals the area of the parallelogram spanned by a and b. The direction follows the right-hand rule.

**Verification**: (3,0,0) x (0,2,0) = (0*0-0*2, 0*0-3*0, 3*2-0*0) = (0, 0, 6). Dot product with a: (0,0,6)·(3,0,0) = 0. Dot product with b: (0,0,6)·(0,2,0) = 0. Perpendicularity confirmed by numbers.

Formula

Formula

a = (a1, a2, a3), b = (b1, b2, b3) a x b = ( a2*b3 - a3*b2, <- X-component a3*b1 - a1*b3, <- Y-component a1*b2 - a2*b1 <- Z-component ) MNEMONIC (3x3 determinant): | i j k | a x b = | a1 a2 a3 | | b1 b2 b3 | PROPERTIES: |a x b| = |a| * |b| * sin(theta) <- parallelogram area a x b = -(b x a) <- anti-commutativity a x a = 0 <- zero vector (sin(0) = 0) 3D ONLY: in 2D there is no third dimension for the result.

**Anti-commutativity is a common trap**: a x b = -(b x a). With the dot product, order does not matter. With the cross product, swapping the order flips the sign of the result. If a normal points the wrong way in a 3D engine, it is almost always a wrong operand order.

Right-hand rule: direction of the result

Right-hand rule: direction of the result

The direction of a x b follows the **right-hand rule**: point the index finger along a, the middle finger along b, and the thumb points in the direction of a x b.

Standard unit vectors: i=(1,0,0), j=(0,1,0), k=(0,0,1) i x j = k (X x Y = Z) <- right-hand rule j x k = i (Y x Z = X) k x i = j (Z x X = Y) and in reverse: j x i = -k (order changed - sign flipped) k x j = -i i x k = -j To remember: i -> j -> k -> i -> ... cyclic = positive. Against the cycle = negative.

Application #1: lighting in 3D graphics (GPU)

Application #1: lighting in 3D graphics (GPU)

**Every triangle in a 3D scene rendered by Unreal Engine, Unity, or WebGL has a normal.** A normal is a vector perpendicular to the surface. The GPU uses it to compute how much light reaches the surface: the angle between the normal and the light direction gives brightness via the dot product. The normal is built from the cross product of the triangle's edges.

**Scale**: a single frame in a game like Cyberpunk 2077 contains hundreds of thousands of triangles. The GPU computes the normal and lighting for each one in parallel within milliseconds. One frame = hundreds of millions of cross product operations. That is why graphics cards are specialized vector processors.

Application #2: torque and angular velocity

Application #2: torque and angular velocity

In physics and robotics, **torque** is tau = r x F, where r is the position vector and F is the force. The result is perpendicular to both - it is the axis of rotation. Boston Dynamics robots, DJI drones, and KUKA manipulators apply this formula at every control step.

EXAMPLE: a wrench. Wrench length r = (0.2, 0.0, 0.0) meters Applied force F = (0.0, 50.0, 0.0) Newtons tau = r x F = (0.2, 0, 0) x (0, 50, 0) = (0*0 - 0*50, 0*0 - 0.2*0, 0.2*50 - 0*0) = (0, 0, 10) N*m Torque = 10 Nm along the Z axis - the bolt's rotation axis. |tau| = |r| * |F| * sin(90) = 0.2 * 50 * 1 = 10 Nm Physical meaning: the longer the wrench (|r|), the greater the torque for the same applied force.

Application #3: camera orientation in SLAM

Application #3: camera orientation in SLAM

**SLAM** (Simultaneous Localization and Mapping) is the algorithm that robots and self-driving cars use to build a map and locate themselves within it. A camera is described by three vectors: forward (where it looks), up (scene up direction), and right. The right vector = forward x up. This is cross product in a real production system.

**Gram-Schmidt via cross product**: the pattern forward -> right (cross) -> up (cross) is part of Gram-Schmidt orthogonalization. In NeRF, 3D Gaussian Splatting, and any system requiring a camera coordinate frame - exactly this code.

Triangle area via cross product

Triangle area via cross product

The length of a x b is the area of the parallelogram. A triangle's area is exactly half that. This is used in 3D rasterization to check whether a pixel falls inside a triangle (barycentric coordinates).

PARALLELOGRAM: S = |a x b| = |a| * |b| * sin(theta) TRIANGLE: S_triangle = |a x b| / 2 EXAMPLE: a = (3, 0, 0), b = (1, 2, 0) a x b = (0*0 - 0*2, 0*1 - 3*0, 3*2 - 0*1) = (0, 0, 6) |a x b| = 6 S_parallelogram = 6 S_triangle = 3 VERIFICATION via basic formula: base = |a| = 3 height = |b| * sin(theta) = sqrt(5) * sin(arctan(2)) = 2 S = 3 * 2 / 2 = 3 << matches.

Dot product vs cross product

Dot product vs cross product

Cross product in industry

Three classes of tasks - one operation

Practice: 3D lighting

Practice: 3D lighting

Interview questions

Key takeaways

• **a x b** - vector perpendicular to both; |a x b| = parallelogram area
• **3D only**: the formula requires exactly three components
• **Anti-commutativity**: a x b = -(b x a); order determines the normal direction
• **GPU rendering**: triangle normal = cross product of two edges
• **Robotics**: torque tau = r x F; the result is the rotation axis
• **SLAM / NeRF**: right = forward x up builds the camera coordinate frame
• **numpy.cross(a, b)** - the whole operation in one line

What comes next

The cross product unlocks the geometry of 3D space

• Lines and planes — A plane's normal is built via the cross product of its edges
• Determinant — The cross product formula is the expansion of a 3x3 determinant
• Rotation matrices — Three vectors (right, up, forward) assemble into a camera rotation matrix

Связанные уроки

• calc-19-gradient