Vectors in 2D and 3D

Vector Notation and Representation

A vector can be written as:

• A bold letter: a (in print) or with an arrow: $\vec{a}$ (in handwriting, use the arrow notation)
• A column vector: $\vec{a} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix}$ in 2D or $\vec{a} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}$ in 3D
• In terms of unit vectors: $\vec{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k}$

A vector from point $A$ to point $B$ is written $\overrightarrow{AB} = B - A$ (position vector of $B$ minus position vector of $A$).

Magnitude of a Vector

The magnitude (length) of $\vec{a} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}$ is:

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

In 2D, simply omit the $a_3$ term. A unit vector has magnitude 1. The unit vector in the direction of $\vec{a}$ is $\hat{a} = \frac{\vec{a}}{|\vec{a}|}$.

Vector Operations

Addition and Subtraction

$\begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} + \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix} = \begin{pmatrix} a_1 + b_1 \\ a_2 + b_2 \\ a_3 + b_3 \end{pmatrix}$

Geometrically, vector addition follows the triangle rule or parallelogram rule. Subtraction $\vec{a} - \vec{b}$ is equivalent to $\vec{a} + (-\vec{b})$.

Scalar Multiplication

$k\vec{a} = k\begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} = \begin{pmatrix} ka_1 \\ ka_2 \\ ka_3 \end{pmatrix}$

Multiplying by a scalar $k$ scales the magnitude by $|k|$ and reverses the direction if $k < 0$.

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

The Scalar (Dot) Product

The scalar product (also called the dot product) of two vectors is defined in two equivalent ways:

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

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

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

Both forms are in your IB formula booklet. The key properties:

• The result is a scalar (a number), not a vector
• $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$ (commutative)
• If $\vec{a} \cdot \vec{b} = 0$ and neither vector is the zero vector, then $\vec{a}$ and $\vec{b}$ are perpendicular
• $\vec{a} \cdot \vec{a} = |\vec{a}|^2$

Angle Between Two Vectors

By rearranging the geometric definition:

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

This is one of the most frequently tested formulas in the IB. The angle $\theta$ always satisfies $0° \leq \theta \leq 180°$.

Steps to find the angle:

1. Calculate $\vec{a} \cdot \vec{b}$ using the algebraic formula
2. Calculate $|\vec{a}|$ and $|\vec{b}|$
3. Substitute into the formula and solve for $\theta$ using $\cos^{-1}$

Perpendicular and Parallel Vectors

• Perpendicular: $\vec{a} \perp \vec{b} \iff \vec{a} \cdot \vec{b} = 0$
• Parallel: $\vec{a} \parallel \vec{b} \iff \vec{b} = k\vec{a}$ for some scalar $k$

These conditions are used frequently in IB problems.

Vector Equation of a Line

A line in 2D or 3D can be written in vector form:

$\vec{r} = \vec{a} + t\vec{d}$

where:

• $\vec{r}$ is the position vector of any point on the line
• $\vec{a}$ is the position vector of a known point on the line
• $\vec{d}$ is the direction vector of the line
• $t$ is a scalar parameter ($t \in \mathbb{R}$)

In component form for 3D: $\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} + t\begin{pmatrix} d_1 \\ d_2 \\ d_3 \end{pmatrix}$

This gives parametric equations: $x = a_1 + td_1$, $y = a_2 + td_2$, $z = a_3 + td_3$.

Finding the Direction Vector

If you know two points $A$ and $B$ on the line, the direction vector is $\vec{d} = \overrightarrow{AB} = B - A$. Note that any scalar multiple of $\vec{d}$ is also a valid direction vector.

Intersection of Lines

To check whether two lines intersect:

1. Write both lines in parametric form (using different parameters, e.g., $t$ and $s$)
2. Set the position vectors equal: $\vec{a_1} + t\vec{d_1} = \vec{a_2} + s\vec{d_2}$
3. This gives a system of equations — solve for $t$ and $s$
4. If a consistent solution exists, the lines intersect (substitute to find the point)
5. If no consistent solution exists, the lines do not intersect (they are skew in 3D, or parallel)

In 3D, two non-parallel lines may be skew — they do not intersect and are not parallel. This cannot happen in 2D.

Distance Concepts

The distance between two points $A$ and $B$ is $|\overrightarrow{AB}|$.

At HL, you may also need to find the shortest distance from a point to a line, which involves projection concepts.

Tip 1: Draw a Diagram

For vector problems, always sketch the situation — even a rough diagram. Mark known points, vectors, and the unknown you are trying to find. This helps you identify which vectors to add or subtract.

Tip 2: Use Position Vectors Carefully

$\overrightarrow{AB} = \vec{b} - \vec{a}$ (end minus start). A common error is computing $\vec{a} - \vec{b}$ instead, which gives the vector pointing in the opposite direction.

Tip 3: Check Perpendicularity with the Dot Product

Whenever a question involves right angles or perpendicularity, immediately think "dot product equals zero." This is the standard approach in IB vector problems.

Tip 4: Use Different Parameters for Different Lines

When finding the intersection of two lines, use different letters (e.g., $t$ and $s$) for the parameters. Using the same letter implies the points are reached at the same "time," which may not be the case.

Tip 5: Verify Your Intersection Point

After finding values of $t$ and $s$, substitute into ALL equations (including any third equation in 3D) to verify the solution is consistent. In 3D, if two equations give valid $t$ and $s$ but the third does not match, the lines are skew.