Cross Product

The cross product is the second fundamental vector operation in electromagnetics and physics. Unlike the dot product which gives a scalar, the cross product of two vectors gives a new vector perpendicular to both input vectors. This property makes it indispensable for computing torque, angular momentum, magnetic force on a moving charge, and the direction of wave propagation in electromagnetic fields.

Core Concept Explanation

The cross product of vectors A and B is defined as A cross B = |A||B|sin(theta) times the unit normal vector n-hat, where n-hat points in the direction determined by the right-hand rule. Curl the fingers of the right hand from A toward B through the smaller angle theta, and the extended thumb points in the direction of A cross B. This physical rule is critical for determining the direction of magnetic force (F = qv cross B) and torque (tau = r cross F).

Unlike the dot product, the cross product is not commutative. In fact, B cross A = negative of A cross B. This anti-commutativity is a fundamental property. If you reverse the order of multiplication, the direction of the result flips by 180 degrees. This matters enormously in problems involving magnetic force where the sign of charge and direction of motion both affect the direction of deflection.

When two vectors are parallel (theta = 0 or 180 degrees), sin(theta) = 0 and the cross product is the zero vector. This makes physical sense: a charge moving parallel to a magnetic field experiences no magnetic force. When two vectors are perpendicular, sin(90) = 1 and the cross product achieves its maximum magnitude of |A||B|.

Mathematical Expression

In Cartesian coordinates, the cross product is evaluated using the 3x3 determinant with unit vectors ax, ay, az in the first row, components of A in the second row, and components of B in the third row. Expanding this determinant gives: A cross B = (AyBz minus AzBy) ax + (AzBx minus AxBz) ay + (AxBy minus AyBx) az. This formula must be memorized for GATE since many problems give component forms directly.

The unit vector cross product rules follow the cyclic order: ax cross ay = az, ay cross az = ax, az cross ax = ay. Reversing any pair gives the negative unit vector: ay cross ax = negative az. A cross A = 0 always, since sin(0) = 0. These rules are derived from the determinant and must be internalized to solve problems quickly without recomputing the full determinant every time.

Practical Understanding

In EM, the cross product is ubiquitous. The magnetic force on a moving charge is F = q(v cross B), where v is the velocity of the charge and B is the magnetic flux density. The force is perpendicular to both the velocity and the field, which is why magnetic fields do no work on charges: the force is always perpendicular to motion. The Poynting vector S = E cross H gives the direction and density of power flow in an electromagnetic wave, and requires the cross product to identify the propagation direction from the electric and magnetic field directions.

The magnitude of the cross product, |A||B|sin(theta), equals the area of the parallelogram formed by A and B as adjacent sides. This geometric interpretation is used in computing surface areas and in defining surface integrals in vector calculus.

Given:
A = 2ax + 3ay + 0az
B = 1ax + 0ay + 4az

Why this formula applies:
Components are given in Cartesian form; use determinant expansion directly.

Formula:
A × B = (AyBz - AzBy)ax + (AzBx - AxBz)ay + (AxBy - AyBx)az

Substitution:
ax component: (3×4) - (0×0) = 12
ay component: (0×1) - (2×4) = -8
az component: (2×0) - (3×1) = -3

Calculation:
A × B = 12ax - 8ay - 3az

|A × B| = sqrt(12² + (-8)² + (-3)²)
        = sqrt(144 + 64 + 9)
        = sqrt(217) ≈ 14.73

Final Answer:
A × B = 12ax - 8ay - 3az (vector), Magnitude ≈ 14.73 units²

Exam Tip: GATE frequently tests the direction of A × B using anti-commutativity. Remember A × B = -(B × A). Also, if a problem asks for a unit vector perpendicular to both A and B, compute A × B first, then divide by its magnitude. This is a two-step process many students forget.

Key Points on Cross Product Mechanism

• Cross product produces a vector, not a scalar. Its direction is perpendicular to both input vectors.
• Direction is given by the right-hand rule: curl from A to B, thumb gives A cross B direction.
• Anti-commutativity: B cross A = negative (A cross B). Order matters, unlike the dot product.
• When vectors are parallel, cross product = 0. Maximum when they are perpendicular.
• Determinant method is the standard computational approach for Cartesian components.
• In EM: magnetic force F = q(v cross B), Poynting vector S = E cross H, torque tau = r cross F all use cross product.

Quick Revision

• A cross B = |A||B|sin(theta) n-hat. Result is a VECTOR perpendicular to both A and B.
• Evaluate using 3x3 determinant with rows: [ax ay az], [Ax Ay Az], [Bx By Bz].
• Unit vector rules: ax cross ay = az (cyclic). Reverse order negates the result.
• A cross A = 0. Parallel vectors always give zero cross product.
• Anti-commutative: A cross B = -(B cross A). This is NOT commutative.
• GATE applications: magnetic force (F = qv cross B), Poynting vector (S = E cross H).
• GATE trap: computing ay component sign. Remember it is +(AzBx minus AxBz) from the cofactor expansion, not the straightforward product.