Right-Hand Rule

Cross product basics

The cross product A × B is a vector perpendicular to both A and B, with magnitude:

|A × B| = |A| · |B| · sin θ

where θ is the angle between them. Direction given by right-hand rule.

For unit vectors: i × j = k, j × k = i, k × i = j (right-handed coordinate system).

Right-hand rule applications

Equation	Hand position	Result
F = qv × B	Fingers along v, curl to B	Thumb = F (for + q)
B around wire	Thumb along current	Fingers = B direction
Solenoid B	Fingers curl in current direction	Thumb = N pole
τ = r × F	Fingers along r, curl to F	Thumb = τ
L = r × p	Fingers along r, curl to p	Thumb = L
Coriolis (in rotating frame)	Fingers along ω, curl to v	Thumb = -F_C / 2m direction

Worked examples

Force on positive charge moving north in upward B field

v = north (let's say +y). B = up (+z). Right-hand: fingers point north, curl up — thumb points east (+x). Force on POSITIVE charge is east.

For an electron (negative), reverse: force is WEST.

Magnetic field around a wire pointing toward you

Conventional current toward you (out of page). Thumb out of page. Fingers curl COUNTERCLOCKWISE around the wire (viewed looking at it).

Solenoid polarity

Wrap right hand around solenoid with fingers pointing in current direction. Thumb points to NORTH pole. Field inside solenoid points from S to N (same direction as your thumb).

JavaScript — cross products and right-hand rule

// Cross product (3D vectors)
function crossProduct(A, B) {
  return [
    A[1]*B[2] - A[2]*B[1],
    A[2]*B[0] - A[0]*B[2],
    A[0]*B[1] - A[1]*B[0]
  ];
}

// Magnitude
function magnitude(v) {
  return Math.sqrt(v[0]*v[0] + v[1]*v[1] + v[2]*v[2]);
}

// Test: i × j should = k
console.log(crossProduct([1,0,0], [0,1,0]));  // [0, 0, 1] = k ✓
console.log(crossProduct([0,1,0], [1,0,0]));  // [0, 0, -1] = -k

// Force on charge: F = q · (v × B)
function lorentzForce(q, v, B) {
  return crossProduct(v, B).map(c => q * c);
}

// Electron (charge -e) moving in +x with B in +z
const e = -1.602e-19;
console.log(lorentzForce(e, [1e6, 0, 0], [0, 0, 1]));
// [0, +1.6e-13, 0] — force is +y direction (right of motion in this frame)

// For positive charge same setup, force would be -y

// Right-hand rule for solenoid: given current direction, find pole
function solenoidNorthPole(currentDirection) {
  // currentDirection: 'clockwise viewed from one end' or 'counterclockwise'
  // Right-hand wraps fingers in current direction; thumb points to N
  // If current is counterclockwise when viewed from end A, then A is N
  return currentDirection === 'counterclockwise' ? 'this end is N' : 'this end is S';
}

// Torque (r × F)
function torque(r, F) {
  return crossProduct(r, F);
}

// 1 m lever arm in +x, 10 N force in +y
console.log(`Torque: ${torque([1, 0, 0], [0, 10, 0])}`);  // [0, 0, 10]
// Thumb: +z (counterclockwise looking from +z)

// Angular momentum (r × p)
function angularMomentum(r, p) {
  return crossProduct(r, p);
}

// Particle moving in circle: r in +x, p in +y, L should be +z
console.log(angularMomentum([0.1, 0, 0], [0, 1, 0]));  // [0, 0, 0.1]

// Verify orthogonality
function isOrthogonal(v1, v2) {
  const dot = v1[0]*v2[0] + v1[1]*v2[1] + v1[2]*v2[2];
  return Math.abs(dot) < 1e-9;
}

const a = [3, 1, 2];
const b = [1, -1, 0];
const cross = crossProduct(a, b);
console.log(`Cross perpendicular to a: ${isOrthogonal(cross, a)}`); // true
console.log(`Cross perpendicular to b: ${isOrthogonal(cross, b)}`); // true

Where the right-hand rule matters

• Electromagnetism throughout. Forces, fields, EMF directions all involve right-hand rule.
• Mechanics. Torque and angular momentum directions.
• Coordinate systems. All standard (x, y, z) systems are right-handed.
• Engineering. Motor and generator design, antenna polarization, photography (3-axis stabilization).
• Computer graphics. Cross products for surface normals, lighting calculations.
• Robotics. Joint rotations, end-effector orientation, grasping.
• Education. Universal mnemonic in EM courses; key tool for visualizing 3D vector relationships.

Common mistakes

• Using left hand. Always RIGHT hand for standard physics. Some special textbooks use Fleming's "left-hand rule" for motors — different mnemonic, same result.
• Forgetting sign for negative charges. v × B gives force direction for positive charge; reverse for electrons.
• Curling wrong way. Always curl from FIRST vector to SECOND (taking shorter rotation). Direction of curl + thumb gives result.
• Mixing up vectors. Identify which is "r" or "v" (first) and which is "F" or "B" (second). Order matters in cross product.
• 3D visualization. Hard to do without practice. Drawing diagrams helps.
• Treating it as commutative. A × B = -(B × A). Order matters.