Vector Addition and Subtraction

Vector addition and subtraction form the foundation of all electromagnetic field analysis. Electric fields from multiple charges, magnetic fields from several current sources, and force components on charges in combined fields are all computed by adding or subtracting vector quantities. Without a clear understanding of how vectors combine, it is impossible to solve problems involving superposition of fields or resultant forces.

In GATE electromagnetics, vector addition appears in questions on resultant electric field due to multiple point charges, net force on a charge, and combined field components in Cartesian, cylindrical, or spherical coordinates.

Core Concept Explanation

A vector quantity has both magnitude and direction, unlike a scalar which has only magnitude. Electric field intensity E, magnetic flux density B, and force F are all vectors. When two or more vectors act at the same point, the net effect is found by vector addition, not scalar addition. Simply adding magnitudes without considering direction gives incorrect results.

The parallelogram law states that if two vectors A and B act at a point, they can be represented as adjacent sides of a parallelogram, and the diagonal of the parallelogram represents their vector sum. Equivalently, in the head-to-tail method, B is placed so its tail touches the head of A, and the resultant vector goes from the tail of A to the head of B.

In practice, vector addition is always performed using components. In Cartesian coordinates, only components along the same axis can be directly added. The x-component of the resultant is the sum of the x-components of all individual vectors, and similarly for y and z. This component-wise approach removes the need for geometric construction and works in all three dimensions.

Vector subtraction A - B is treated as adding A and the negative of B. The vector -B has the same magnitude as B but points in exactly the opposite direction. This concept is important in computing the displacement vector between two points, which appears in Coulomb's law and in finding the distance vector r12.

Mathematical Expression

Given two vectors A = Ax ax + Ay ay + Az az and B = Bx ax + By ay + Bz az, the sum and difference are:

A + B = (Ax + Bx) ax + (Ay + By) ay + (Az + Bz) az

A - B = (Ax - Bx) ax + (Ay - By) ay + (Az - Bz) az

The magnitude of the resultant vector C = A + B is |C| = sqrt(Cx^2 + Cy^2 + Cz^2). The unit vector in the direction of the resultant is aC = C / |C|. This unit vector is needed when expressing a force or field in terms of its direction.

For vectors in cylindrical or spherical coordinates, direct component-wise addition is valid only if both vectors are expressed at the same point. Since unit vectors in these systems are position-dependent, vectors at different points must first be converted to Cartesian, then added, then converted back if needed.

Practical Understanding

The superposition principle in electromagnetics states that the total electric field at a point due to multiple charges equals the vector sum of the individual fields. For two charges Q1 and Q2, the resultant field E = E1 + E2 is computed by first finding E1 and E2 individually in Cartesian component form, then adding the corresponding components.

In force analysis, a charge Q in a region with both electric field E and magnetic field B experiences the Lorentz force F = Q(E + v x B). Computing this requires vector addition of the electric force QE and the magnetic force Q(v x B), both of which are vectors in three dimensions.

Given:
Two electric field vectors at point P:
E1 = 3 ax + 4 ay + 0 az  V/m
E2 = 1 ax + 2 ay + 6 az  V/m
Find: resultant field E, its magnitude, and unit vector.

Why this formula applies:
Superposition principle: total E = vector sum of all individual fields.
Component-wise addition applies since both are in Cartesian at same point.

Formula:
E = E1 + E2 = (E1x+E2x)ax + (E1y+E2y)ay + (E1z+E2z)az
|E| = sqrt(Ex^2 + Ey^2 + Ez^2)
aE = E / |E|

Substitution:
Ex = 3 + 1 = 4
Ey = 4 + 2 = 6
Ez = 0 + 6 = 6

Calculation:
E = 4 ax + 6 ay + 6 az  V/m
|E| = sqrt(4^2 + 6^2 + 6^2)
|E| = sqrt(16 + 36 + 36) = sqrt(88) = 9.38 V/m

aE = (4 ax + 6 ay + 6 az) / 9.38
aE = 0.426 ax + 0.639 ay + 0.639 az

Verification:
|aE| = sqrt(0.426^2 + 0.639^2 + 0.639^2) = sqrt(0.182 + 0.408 + 0.408) = sqrt(0.998) ≈ 1

Final Answer:
Resultant field E = 4 ax + 6 ay + 6 az  V/m
Magnitude |E| = 9.38 V/m
Unit vector aE = 0.426 ax + 0.639 ay + 0.639 az

Exam Tip: In GATE, when adding vectors in cylindrical or spherical coordinates from different points in space, you must first convert all vectors to Cartesian before adding. Direct addition of arho or ar components from different points is incorrect because those unit vectors point in different directions at different locations.

• Vector addition requires adding components along the same axis. Components along different axes cannot be combined.
• Parallelogram law and head-to-tail method both give the same resultant and are geometrically equivalent.
• Subtraction: A - B equals A plus (-B). The vector -B has identical magnitude to B but reversed direction.
• Superposition principle: resultant field is the vector sum of all individual fields. This is valid for electric and magnetic fields in linear media.
• For cylindrical or spherical vectors at different points, conversion to Cartesian is required before addition because position-dependent unit vectors do not stay constant between points.

Quick Revision

• Addition: C = A + B. Components: Cx = Ax+Bx, Cy = Ay+By, Cz = Az+Bz.
• Subtraction: C = A - B. Components: Cx = Ax-Bx, Cy = Ay-By, Cz = Az-Bz.
• Magnitude: |C| = sqrt(Cx^2 + Cy^2 + Cz^2).
• Unit vector of resultant: aC = C / |C|. Always verify |aC| = 1.
• Superposition applies to E and B fields: total field = vector sum of individual contributions.
• Parallelogram law: diagonal of parallelogram formed by A and B equals A + B.
• Key GATE trap: never add vectors in non-Cartesian coordinates if they are from different spatial locations. Convert to Cartesian first.