Vectors

A vector has both magnitude and direction. A scalar has magnitude only. Velocity, displacement, and force are vectors; speed, mass, and temperature are scalars.

Notation¶

Vectors are written in bold (a) in typeset text, or underlined ($\underline{a}$) in handwriting. In two dimensions, a vector is expressed as a column vector

$\mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix}$

or in terms of the standard unit vectors $\mathbf{i} = \begin{pmatrix}1\\0\end{pmatrix}$ and $\mathbf{j} = \begin{pmatrix}0\\1\end{pmatrix}$ as $\mathbf{v} = x\mathbf{i} + y\mathbf{j}$.

A position vector $\overrightarrow{OP}$ describes the location of point $P$ relative to the origin $O$.

Operations on Vectors¶

Addition and subtraction: add or subtract corresponding components.

$\begin{pmatrix}a\\b\end{pmatrix} + \begin{pmatrix}c\\d\end{pmatrix} = \begin{pmatrix}a+c\\b+d\end{pmatrix}$

Scalar multiplication: multiply each component by the scalar $k$. This scales the magnitude by $|k|$ and reverses direction if $k < 0$.

$k\begin{pmatrix}a\\b\end{pmatrix} = \begin{pmatrix}ka\\kb\end{pmatrix}$

Equal vectors have the same magnitude and direction; position does not matter.

Magnitude¶

The magnitude (length) of $\mathbf{v} = \begin{pmatrix}x\\y\end{pmatrix}$ is:

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

This is Pythagoras applied to the horizontal and vertical components.

Direction of a Vector¶

The direction of a vector is the angle it makes with the positive $x$-axis, measured anticlockwise. For a vector with components $(x, y)$:

$\theta = \arctan\left(\frac{y}{x}\right)$

$\arctan$ returns values between $-90^\circ$ and $90^\circ$, so the quadrant must be checked. For a vector in Quadrant II or III, add $180^\circ$ to the result.

Unit Vectors¶

A unit vector has magnitude 1. To find the unit vector in the direction of $\mathbf{a}$:

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

Displacement Vectors¶

The vector from point $A$ to point $B$ is:

$\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA} = \mathbf{b} - \mathbf{a}$

Equivalently, $\overrightarrow{AB} = B - A$ (coordinates of $B$ minus coordinates of $A$). The order matters: $\overrightarrow{AB} = -\overrightarrow{BA}$.

Parallel and Collinear Vectors¶

Two vectors are parallel if one is a scalar multiple of the other: $\mathbf{b} = k\mathbf{a}$ for some scalar $k$.

Three points $A$, $B$, $C$ are collinear if $\overrightarrow{AB}$ is parallel to $\overrightarrow{AC}$ (and both share point $A$).

The Scalar (Dot) Product¶

The scalar product of $\mathbf{a} = \begin{pmatrix}a_1\\a_2\end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix}b_1\\b_2\end{pmatrix}$ is defined two ways:

Component form:

$\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2$

Geometric form:

$\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos\theta$

where $\theta$ is the angle between the vectors ($0 \leq \theta \leq \pi$).

The result is a scalar (a number), not a vector.

Finding the Angle Between Two Vectors¶

Combining both definitions:

$\cos\theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|}$

Perpendicular Vectors¶

Two non-zero vectors are perpendicular if and only if their dot product is zero:

$\mathbf{a} \cdot \mathbf{b} = 0 \iff \mathbf{a} \perp \mathbf{b}$

This follows directly from the geometric form: $\cos 90° = 0$.

Properties of the Scalar Product¶

• Commutative: $\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}$
• $\mathbf{a} \cdot \mathbf{a} = |\mathbf{a}|^2$ (useful for finding $|\mathbf{a}|$ without the square root formula when the vector is expressed algebraically)
• Distributive: $\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}$ (used when expanding expressions involving sums of vectors)

Scalars vs Vectors: Common Examples¶