Vectors: Fundamentals

Vector Notation

A vector is a quantity with both magnitude and direction. Common notations:

• Bold Roman: $\mathbf{u}$, $\mathbf{q}$ (typeset)
• Underline/tilde: $\underline{u}$ (handwritten)
• Arrow: $\vec{u}$ (North American convention)

A position vector from point $A$ to point $B$ is written $\vec{AB}$. Position vectors from the origin $O$ to point $A$ are written $\vec{OA}$ or simply $\mathbf{a}$.

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Vector Addition

Vectors add geometrically using the parallelogram rule or triangle rule:

$\mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a} \quad \text{(commutative)}$

$(\mathbf{a} + \mathbf{b}) + \mathbf{c} = \mathbf{a} + (\mathbf{b} + \mathbf{c}) \quad \text{(associative)}$

For three non-coplanar vectors, their sum is the diagonal of a parallelepiped formed by the vectors as edges.

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Scalar Multiplication

Multiplying a vector $\mathbf{a}$ by a scalar $\lambda \in \mathbb{R}$:

$\lambda(\mathbf{a} + \mathbf{b}) = \lambda\mathbf{a} + \lambda\mathbf{b} \quad \text{(distributive)}$

• $\lambda > 0$: stretches/compresses without changing direction
• $\lambda < 0$: reverses direction and scales magnitude
• $|\lambda\mathbf{a}| = |\lambda||\mathbf{a}|$

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Cartesian Components

Right-Handed Coordinate System

By convention, we use a right-handed system where the $x$, $y$, $z$ axes satisfy:

• Thumb points along $x$
• Index finger along $y$
• Middle finger along $z$

Unit Vectors

The standard basis (orthonormal set):

$\hat{\mathbf{i}}, \quad \hat{\mathbf{j}}, \quad \hat{\mathbf{k}} \quad \text{or} \quad \mathbf{e}_x, \mathbf{e}_y, \mathbf{e}_z$

Properties:

• Orthogonal: $\hat{\mathbf{i}} \cdot \hat{\mathbf{j}} = \hat{\mathbf{j}} \cdot \hat{\mathbf{k}} = \hat{\mathbf{k}} \cdot \hat{\mathbf{i}} = 0$
• Unit length: $|\hat{\mathbf{i}}| = |\hat{\mathbf{j}}| = |\hat{\mathbf{k}}| = 1$

Position Vector

$\mathbf{r} = x\hat{\mathbf{i}} + y\hat{\mathbf{j}} + z\hat{\mathbf{k}} = (x, y, z)$

Magnitude (Length)

$|\mathbf{r}| = \sqrt{x^2 + y^2 + z^2}$

Distance Between Points

$|\mathbf{r}_2 - \mathbf{r}_1| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

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Unit Vectors and Direction

A unit vector has magnitude 1. To normalise any non-zero vector $\mathbf{a}$:

$\hat{\mathbf{a}} = \frac{\mathbf{a}}{|\mathbf{a}|}$

This extracts the direction while discarding magnitude information.

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Why Vectors Matter

Vectors unify many physical quantities:

• Displacement: position change
• Velocity: rate of position change
• Acceleration: rate of velocity change
• Force: causes acceleration
• Electric/magnetic fields: act on charges

The operations (addition, scalar multiplication) match physical intuition: displacements add, forces combine, velocities scale.