Vector Algebra

A scalar is a quantity described completely by a single number (with its unit): temperature, mass, electric charge, time, or electric potential. A vector needs both a magnitude and a direction: force, velocity, and — at the heart of this course — the electric field \(\vec{E}\) and the magnetic field \(\vec{H}\).

In electromagnetics a field is a quantity defined at every point of a region. A scalar field, such as the potential \(V(x,y,z)\), assigns a number to each point; a vector field, such as \(\vec{E}(x,y,z)\), assigns an arrow — a magnitude and direction — to each point. Mastering the algebra of a single vector is the prerequisite for handling fields made of infinitely many of them.

We write a vector in boldface or with an arrow, \(\vec{A}\), and its magnitude as \(A = |\vec{A}|\). A vector of magnitude one is a unit vector, written \(\hat{a}\).

In the Cartesian system, three mutually perpendicular unit vectors \(\hat{a}_x,\ \hat{a}_y,\ \hat{a}_z\) point along the \(x\), \(y\) and \(z\) axes. They form a right-handed set: curling the fingers of the right hand from \(\hat{a}_x\) to \(\hat{a}_y\) makes the thumb point along \(\hat{a}_z\). Any vector \(\vec{A}\) is the sum of its projections on these three axes:

The numbers \(A_x, A_y, A_z\) are the components (or scalar components) of \(\vec{A}\). Because the axes are perpendicular, the magnitude follows from the three-dimensional Pythagoras theorem, and the unit vector divides the whole thing by that length:

Vectors add component by component. Geometrically this is the parallelogram (or head-to-tail) rule: place the tail of \(\vec{B}\) at the head of \(\vec{A}\), and the resultant \(\vec{C}\) runs from the start of \(\vec{A}\) to the end of \(\vec{B}\).

Addition is commutative (\(\vec{A}+\vec{B}=\vec{B}+\vec{A}\)) and associative. Multiplying by a scalar \(k\) scales every component, stretching the vector and — if \(k\) is negative — reversing it.

The position vector \(\vec{r}_P\) of a point \(P(x,y,z)\) runs from the origin to \(P\):

The distance vector (or separation vector) from point \(P\) to point \(Q\) is the difference of their position vectors. It is the single most-used object in field theory: in Coulomb's law and the Biot–Savart law, the field at \(Q\) depends on the distance vector from the source at \(P\).

The dot (scalar) product multiplies two vectors to give a scalar. Geometrically it measures how much one vector lies along another — a projection:

Key consequences: the dot product is commutative; it is zero when the vectors are perpendicular (\(\cos 90^\circ = 0\)); and \(\vec{A}\cdot\vec{A} = |\vec{A}|^2\). For the base vectors, \(\hat{a}_x\cdot\hat{a}_x = 1\) and \(\hat{a}_x\cdot\hat{a}_y = 0\). In electromagnetics the dot product is how we compute work (\(\vec{F}\cdot\vec{L}\)) and flux (\(\vec{D}\cdot\vec{S}\)).

The cross (vector) product multiplies two vectors to give a third vector, perpendicular to both, with magnitude equal to the area of the parallelogram they span:

The unit vector \(\hat{a}_n\) is fixed by the right-hand rule: point the fingers along \(\vec{A}\), curl them toward \(\vec{B}\), and the thumb gives \(\hat{a}_n\). In components the cross product is the determinant:

The cross product is anti-commutative: \(\vec{A}\times\vec{B} = -\,\vec{B}\times\vec{A}\). It is zero for parallel vectors, and \(\hat{a}_x\times\hat{a}_y = \hat{a}_z\) (cyclically). It powers the magnetic force \(\vec{F} = q\vec{v}\times\vec{B}\), torque, and the curl operator we meet in Chapter 3.

Three vectors combine in two useful ways. The scalar triple product returns a number equal to the volume of the parallelepiped they span:

Its value is unchanged by cyclic rotation of the vectors: \(\vec{A}\cdot(\vec{B}\times\vec{C}) = \vec{B}\cdot(\vec{C}\times\vec{A}) = \vec{C}\cdot(\vec{A}\times\vec{B})\). The vector triple product returns a vector and is best remembered by the "bac–cab" rule:

For each item, decide first which operation it tests — magnitude, projection, normal direction, or volume — then apply the matching rule. Difficulty rises down the list.