Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

Vectors are added (or subtracted) component by component:

A \pm B = (A_{x} \pm B_{x}) a_{x} + (A_{y} \pm B_{y}) a_{y} + (A_{z} \pm B_{z}) a_{z} . (2.1.4)

Geometrically this is the familiar head-to-tail (or parallelogram) rule of Figure 2.1.3: walk along $A$ , then along $B$ , and the arrow from the start to the end point is $A + B$ . Addition is commutative, $A + B = B + A$ (the two routes around the parallelogram end at the same corner), and associative, $(A + B) + C = A + (B + C)$ . The difference $A - B$ is $A + (- B)$ . Drawn from a common tail, it is the arrow from the head of $B$ to the head of $A$ , a picture we shall use again for the displacement vector $R$ in Section 2.3.

Multiplying a vector by a scalar $k$ multiplies every component by $k$ , $k A = k A_{x} a_{x} + k A_{y} a_{y} + k A_{z} a_{z}$ . It stretches the vector by the factor $∣ k ∣$ and reverses it if $k < 0$ . Only quantities with the same unit can be added: a force can be added to a force, but not to an electric field.

The geometry of vector addition and scaling. (a) To add \vec B to \vec A, start \vec B at the head of \vec A. (b) The same sum as the diagonal of the parallelogram spanned by \vec A and \vec B. (c) Drawn from a common tail, \vec A-\vec B goes from the head of \vec B to the head of \vec A (check: \vec B+(\vec A-\vec B)=\vec A). (d) k\vec A is parallel to \vec A, |k| times as long, and reversed when k<0. — **Figure 2.1.3.** The geometry of vector addition and scaling. (a) To add $B$ to $A$ , start $B$ at the head of $A$ . (b) The same sum as the diagonal of the parallelogram spanned by $A$ and $B$ . (c) Drawn from a common tail, $A - B$ goes from the head of $B$ to the head of $A$ (check: $B + (A - B) = A$ ). (d) $k A$ is parallel to $A$ , $∣ k ∣$ times as long, and reversed when $k < 0$ .

Remark

Equation (2.1.4) works because $a_{x}, a_{y}, a_{z}$ are the same at every point of space. We shall see in Section 2.3.4 that this is not true of the cylindrical unit vectors $a_{ρ}$ and $a_{ϕ}$ , and that adding cylindrical components of vectors located at different points is a classic mistake.

Common mistake

Adding magnitudes instead of vectors. In general $∣ A + B ∣ \neq = A + B$ : the lengths add only when the two vectors point the same way. For $A = 3 a_{x}$ and $B = 4 a_{y}$ we get $A + B = 3 a_{x} + 4 a_{y}$ with $∣ A + B ∣ = 5$ , not $7$ . Always add the components first and take the magnitude at the end.

2.1.3 Adding, subtracting and scaling vectors