1: Vector algebra

Vector Algebra

Vector algebra provides the foundational operations for manipulating quantities possessing both magnitude and direction, essential for electromagnetism and physics. Unlike scalars (e.g., mass, temperature), vectors (e.g., force, electric field) require specification of direction alongside magnitude.

Key Concepts:

1. Representation: A vector $\mathbf{A}$ in 3D Cartesian coordinates is expressed using unit vectors $(\hat{\imath}, \hat{\jmath}, \hat{k})$ along the $x$, $y$, $z$ axes:
   $\mathbf{A} = A_x \hat{\imath} + A_y \hat{\jmath} + A_z \hat{k}$
   Its magnitude is $|\mathbf{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}$.

2. Addition/Subtraction: Vectors add/subtract component-wise:
   $\mathbf{A} \pm \mathbf{B} = (A_x \pm B_x) \hat{\imath} + (A_y \pm B_y) \hat{\jmath} + (A_z \pm B_z) \hat{k}$
   Geometrically, addition follows the head-to-tail rule. Subtraction is $\mathbf{A} - \mathbf{B} = \mathbf{A} + (-\mathbf{B})$, where $-\mathbf{B}$ reverses direction.

3. Scalar Multiplication: Multiplying a vector by a scalar $c$ scales its magnitude:
   $c\mathbf{A} = (c A_x) \hat{\imath} + (c A_y) \hat{\jmath} + (c A_z) \hat{k}$
   If $c > 0$, direction remains; if $c < 0$, direction reverses.

4. Dot Product (Scalar Product): Measures projection of one vector onto another, yielding a scalar:
   $\mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos \theta = A_x B_x + A_y B_y + A_z B_z$
   Key properties: Commutative ($\mathbf{A} \cdot \mathbf{B} = \mathbf{B} \cdot \mathbf{A}$), Distributive ($\mathbf{A} \cdot (\mathbf{B} + \mathbf{C}) = \mathbf{A} \cdot \mathbf{B} + \mathbf{A} \cdot \mathbf{C}$).
   Orthogonal vectors satisfy $\mathbf{A} \cdot \mathbf{B} = 0$.

5. Cross Product (Vector Product): Produces a vector perpendicular to the plane containing $\mathbf{A}$ and $\mathbf{B}$:
   $\mathbf{A} \times \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \sin \theta \, \hat{\mathbf{n}}$
   $\hat{\mathbf{n}}$ is the unit vector given by the right-hand rule (curl fingers from $\mathbf{A}$ to $\mathbf{B}$; thumb points along $\hat{\mathbf{n}}$).
   Component form:
   $\mathbf{A} \times \mathbf{B} = (A_y B_z - A_z B_y) \hat{\imath} + (A_z B_x - A_x B_z) \hat{\jmath} + (A_x B_y - A_y B_x) \hat{k}$
   Key properties: Anticommutative ($\mathbf{A} \times \mathbf{B} = -\mathbf{B} \times \mathbf{A}$), Distributive.
   Parallel vectors satisfy $\mathbf{A} \times \mathbf{B} = \mathbf{0}$.

6. Triple Products:

   • Scalar Triple Product: $\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C})$ gives the volume of the parallelepiped spanned by the vectors. It's cyclic: $\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) = \mathbf{B} \cdot (\mathbf{C} \times \mathbf{A}) = \mathbf{C} \cdot (\mathbf{A} \times \mathbf{B})$. Vectors are coplanar if $\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) = 0$.
   • Vector Triple Product: $\mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = \mathbf{B} (\mathbf{A} \cdot \mathbf{C}) - \mathbf{C} (\mathbf{A} \cdot \mathbf{B})$ (BAC-CAB rule). Crucial for vector calculus identities.

Mastery of these operations—especially the dot product, cross product, and triple products—is critical for formulating and solving problems in electrostatics, magnetostatics, and electromagnetism.