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

2.1.8 Recap

Recap

A vector $=$ magnitude $\times$ unit vector: $A = A a_{A}$ , with $A = A_{x} a_{x} + A_{y} a_{y} + A_{z} a_{z}$ , $A = A_{x}^{2} + A_{y}^{2} + A_{z}^{2}$ , $a_{A} = A / A$ (dimensionless).
Sums, differences and multiples are formed component by component, or geometrically by the head-to-tail rule. $∣ A + B ∣ \neq = A + B$ in general.
Dot product (a scalar): $A \cdot B = A_{x} B_{x} + A_{y} B_{y} + A_{z} B_{z} = A B cos θ_{A B}$ . It measures a projection, as in $A_{x} = A \cdot a_{x}$ and $d W = F \cdot d ℓ$ , and $A \cdot B = 0 \Leftrightarrow A ⊥ B$ .
Cross product (a vector): the determinant (2.1.8), or $A \times B = a_{n} A B sin θ_{A B}$ (an oriented area, right-hand rule). Order matters: $A \times B = - B \times A$ . Also $a_{x} \times a_{y} = a_{z}$ and cyclically.
Triple products: $A \cdot (B \times C) =$ volume of the parallelepiped (a determinant), and $A \times (B \times C) = B (A \cdot C) - C (A \cdot B)$ .