Vectors

Vector Notation and Representation

A vector can be represented in several ways:

• Column vector: $\mathbf{a} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix}$ in 2D, or $\mathbf{a} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}$ in 3D.
• Component form: $\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j}$ in 2D, or $\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k}$ in 3D, where $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$ are unit vectors along the $x$-, $y$-, and $z$-axes respectively.
• Directed line segment: $\overrightarrow{AB}$ represents the vector from point $A$ to point $B$.

Vectors are typically printed in bold ($\mathbf{a}$) or with an underline ($\underline{a}$) in handwriting. Scalars (ordinary numbers) are not bold.

Position Vectors

The position vector of a point $P$ is the vector $\overrightarrow{OP}$ from the origin $O$ to the point $P$. If $P = (3, -1, 4)$, then the position vector of $P$ is:

$\overrightarrow{OP} = \begin{pmatrix} 3 \\ -1 \\ 4 \end{pmatrix} = 3\mathbf{i} - \mathbf{j} + 4\mathbf{k}$

The vector from point $A$ (with position vector $\mathbf{a}$) to point $B$ (with position vector $\mathbf{b}$) is:

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

This is one of the most frequently used results in vector problems.

Vector Arithmetic

Addition: Vectors are added component by component. $\begin{pmatrix} a_1 \\ a_2 \end{pmatrix} + \begin{pmatrix} b_1 \\ b_2 \end{pmatrix} = \begin{pmatrix} a_1 + b_1 \\ a_2 + b_2 \end{pmatrix}$

Scalar multiplication: Each component is multiplied by the scalar. $\lambda\begin{pmatrix} a_1 \\ a_2 \end{pmatrix} = \begin{pmatrix} \lambda a_1 \\ \lambda a_2 \end{pmatrix}$

Subtraction: $\mathbf{a} - \mathbf{b} = \mathbf{a} + (-\mathbf{b})$.

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

Magnitude of a Vector

The magnitude (or modulus) of a vector $\mathbf{a} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}$ is:

$|\mathbf{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$

In 2D, this simplifies to $|\mathbf{a}| = \sqrt{a_1^2 + a_2^2}$.

The magnitude represents the length of the vector. The distance between points $A$ and $B$ is $|\overrightarrow{AB}|$.

Unit Vectors

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

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

Unit vectors are used to specify direction without regard to magnitude. The standard unit vectors $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$ are unit vectors along the coordinate axes.

The Scalar (Dot) Product

The scalar product (or dot product) of two vectors $\mathbf{a}$ and $\mathbf{b}$ is defined in two equivalent ways:

Algebraic definition: $\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3$

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

where $\theta$ is the angle between the two vectors.

From the geometric definition, the angle between two vectors can be found:

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

Key properties:

• $\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}$ (commutative)
• $\mathbf{a} \cdot \mathbf{a} = |\mathbf{a}|^2$
• If $\mathbf{a} \cdot \mathbf{b} = 0$ and neither vector is the zero vector, then $\mathbf{a}$ and $\mathbf{b}$ are perpendicular.

Midpoints and Section Formulae

The midpoint $M$ of points $A$ and $B$ with position vectors $\mathbf{a}$ and $\mathbf{b}$ has position vector:

$\overrightarrow{OM} = \frac{\mathbf{a} + \mathbf{b}}{2}$

More generally, the point dividing $AB$ in the ratio $m:n$ has position vector:

$\frac{n\mathbf{a} + m\mathbf{b}}{m + n}$

Tip 1: Draw a Diagram

Always start vector problems with a clear diagram showing the points and vectors involved. Label position vectors and direction vectors. This helps you set up the correct vector equations.

Tip 2: Use $\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$

The vector from $A$ to $B$ is the position vector of $B$ minus the position vector of $A$. This is the single most important formula in vector geometry. Getting the order wrong (subtracting the wrong way) is a very common error.

Tip 3: Check Perpendicularity with the Dot Product

To show two vectors are perpendicular, compute their dot product. If $\mathbf{a} \cdot \mathbf{b} = 0$, they are perpendicular. This is cleaner and more efficient than using gradients.

Tip 4: Use Parallel Vectors for Collinearity

To show that three points $A$, $B$, $C$ are collinear (lie on the same straight line), show that $\overrightarrow{AB} = \lambda\overrightarrow{AC}$ for some scalar $\lambda$. This means the vectors are parallel and share a common point.

Tip 5: Be Precise with Notation

Use bold or underlined letters for vectors in your written work. Clearly distinguish between vectors and scalars. Write $|\mathbf{a}|$ for the magnitude, not $\mathbf{|a|}$. Correct notation demonstrates understanding and avoids ambiguity.