Vector Algebra & Coordinate Systems

Electromagnetic fields are inherently vector quantities — they have a magnitude (how strong the field is) and a direction (which way a test charge would be pushed). This means every equation we encounter in this course will involve vectors. Unlike scalar quantities such as temperature or mass, which are fully described by a single number, fields like the electric field E require three components to specify completely.

Fortunately, the vector algebra needed for EM is a small, well-defined toolkit. If you can add vectors, take dot products, take cross products, and switch between coordinate systems, you have the machinery to read and manipulate Maxwell's equations themselves. This lesson refreshes those skills and introduces the three coordinate systems we will use throughout the course.

A vector A in three-dimensional space is written in terms of its components along the three Cartesian axes: A = A_xx̂ + A_yŷ + A_zẑ. Here x̂, ŷ, and ẑ are unit vectors — vectors of length 1 pointing along the x, y, and z axes respectively. The scalars A_x, A_y, A_z are the components. The magnitude (or norm) of A is |A| = √(A_x² + A_y² + A_z²).

Vector addition and subtraction are performed component by component. If A = 2x̂ + 3ŷ + 1ẑ and B = 1x̂ − 1ŷ + 2ẑ, then A + B = (2+1)x̂ + (3−1)ŷ + (1+2)ẑ = 3x̂ + 2ŷ + 3ẑ. Scalar multiplication scales every component: 2A = 4x̂ + 6ŷ + 2ẑ. These operations are straightforward but appear constantly — whenever we superpose fields from multiple sources, we are just adding vectors.

The dot product (also called the scalar product) takes two vectors and returns a single scalar. The geometric definition is A·B = |A||B|cosθ, where θ is the angle between the two vectors. This tells us the dot product measures how much two vectors point in the same direction — it is zero when they are perpendicular and maximized when they are parallel.

In component form, the dot product is far more practical for computation: A·B = A_xB_x + A_yB_y + A_zB_z. Just multiply corresponding components and sum. The dot product is commutative and distributes over addition. In EM, the dot product appears whenever we compute the component of a field along a direction — for instance, the electric flux through a surface depends on E·n̂, the dot product of the field with the surface normal.

The cross product takes two vectors and returns a new vector that is perpendicular to both. Its magnitude is |A×B| = |A||B|sinθ, and the direction is given by the right-hand rule: point your fingers in the direction of A, curl them toward B, and your thumb points in the direction of A×B.

In component form, the cross product is computed as a 3×3 determinant. Expanding along the first row gives: (A_yB_z − A_zB_y)x̂ − (A_xB_z − A_zB_x)ŷ + (A_xB_y − A_yB_x)ẑ. The cross product is not commutative: A×B = −(B×A). In EM, the cross product appears in the Lorentz force law (F = qv×B) and in the definition of the Poynting vector, among many other places.

Many problems in electromagnetics have natural symmetry, and choosing the right coordinate system can turn an intractable problem into a straightforward one. We will use three systems throughout this course.

Cartesian coordinates (x, y, z) are the default; unit vectors x̂, ŷ, ẑ are constant everywhere. Cylindrical coordinates (ρ, φ, z) are ideal for problems with symmetry about an axis — infinitely long wires, coaxial cables, cylindrical capacitors. Here ρ is the radial distance from the z-axis, φ is the azimuthal angle measured from the x-axis, and z is the same as in Cartesian. Spherical coordinates (r, θ, φ) are used for problems with symmetry about a point — point charges, spheres, dipoles. Here r is the distance from the origin, θ is the polar angle measured from the z-axis, and φ is the same azimuthal angle as in cylindrical.

To see how Cartesian-to-cylindrical conversion works in practice, consider the point (x, y, z) = (3, 4, 0). The radial distance is ρ = √(3² + 4²) = √25 = 5. The azimuthal angle is φ = atan2(4, 3) ≈ 53.13°. The z-coordinate carries over unchanged: z = 0. So in cylindrical coordinates, this point is (ρ, φ, z) = (5, 53.13°, 0). Notice that ρ is always non-negative, and atan2 automatically places the angle in the correct quadrant — unlike a simple arctan(y/x), which loses quadrant information.