Vectors and vector algebra (HL)

Q: What is the fundamental difference between a scalar and a vector?

A scalar has only magnitude, whereas a vector has both magnitude and direction .

Q: What are the two common ways to represent a vector in the IB DP curriculum?

The two ways are Column Vector Form , \(\mathbf{v} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}\), and Base Vector Form , \(\mathbf{v} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}\).

Q: How is a position vector defined?

A position vector describes the location of a point relative to a fixed origin \(O(0,0,0)\), written as \(\vec{OP} = \mathbf{p}\).

Q: What is the rule for finding the displacement vector \(\vec{AB}\) between two points?

The rule is "End minus Start": \(\vec{AB} = \mathbf{b} - \mathbf{a}\), where \(\mathbf{a}\) and \(\mathbf{b}\) are the position vectors of points \(A\) and \(B\).

Q: How do you perform vector addition and subtraction algebraically?

You add or subtract the corresponding components of the vectors: \(\mathbf{u} \pm \mathbf{v} = \begin{pmatrix} u_1 \pm v_1 \\ u_2 \pm v_2 \\ u_3 \pm v_3 \end{pmatrix}\).

Q: What happens to a vector when it is multiplied by a scalar \(k\)?

Every component of the vector is multiplied by \(k\). This changes the magnitude and, if \(k\) is negative, reverses the direction.

Q: How do you calculate the magnitude (length) of a vector \(\mathbf{v}\)?

The magnitude is calculated using the 3D Pythagorean distance formula: \(|\mathbf{v}| = \sqrt{x^2 + y^2 + z^2}\).

Q: How is a unit vector in the direction of \(\mathbf{v}\) calculated?

A unit vector \(\hat{\mathbf{v}}\) is found by dividing the vector by its magnitude: \(\hat{\mathbf{v}} = \frac{\mathbf{v}}{|\mathbf{v}|}\).

Q: What is the mathematical condition for two vectors to be parallel?

Two vectors \(\mathbf{u}\) and \(\mathbf{v}\) are parallel if one is a scalar multiple of the other: \(\mathbf{u} = k\mathbf{v}\) for some constant \(k\).

Q: How can vectors be used to prove a quadrilateral \(ABCD\) is a parallelogram?

Show that \(\vec{AB} = \vec{DC}\) (or \(\vec{AD} = \vec{BC}\)).

Mathematics - Analysis and Approaches · IB Diploma Programme (DP) - SL & HL · 6 min read · Updated: 7 Sept 2026

Introduction to Vectors and Vector Algebra (HL)

Welcome to the world of vectors! If you have ever used a GPS to find your way or followed a treasure map ("Walk ten paces North, then five paces East"), you have already used vectors in real life. While a regular number (a scalar) only tells us "how much" (like temperature or mass), a vector tells us "how much" AND "in which direction."

In this chapter, we will learn how to represent vectors mathematically in two and three dimensions, how to perform "vector arithmetic," and how to use these tools to prove geometric theorems. Don't worry if it feels a bit abstract at first; vectors are just a language for describing journeys and positions in space.

Note: This chapter focuses on the foundational algebra of vectors (Syllabus 3.12). You will explore angles and planes in the subsequent chapters of this section.

1. What is a Vector?

A vector is a quantity that has both magnitude (size) and direction. Geometrically, we represent a vector as a directed line segment—basically an arrow starting at one point and ending at another.

Notation and Representation

There are several ways to write a vector. If a vector goes from point \(A\) to point \(B\), we write it as \(\vec{AB}\). If we give the vector a single name, like \(v\), we usually write it in bold (\(\mathbf{v}\)) or with a small underline (\(\underline{v}\)) when writing by hand.

In the IB DP curriculum, we represent vectors using components in two main ways:

Column Vector Form: \(\mathbf{v} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}\)
Base Vector Form: \(\mathbf{v} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}\)

The unit vectors \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) are the standard "building blocks" that point exactly 1 unit in the direction of the \(x\), \(y\), and \(z\) axes respectively.

Quick Tip: Think of a column vector like an instruction set. \(\begin{pmatrix} 3 \\ -2 \\ 5 \end{pmatrix}\) means "Go 3 units right, 2 units down, and 5 units 'up' out of the page."

2. Position vs. Displacement Vectors

It is very important to distinguish between where something is and where it is going.

Position Vectors

A position vector describes the location of a point relative to a fixed origin, \(O(0,0,0)\). If we have a point \(P(x, y, z)\), its position vector is \(\vec{OP} = \mathbf{p} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}\).

Displacement Vectors

A displacement vector describes the "journey" from one point to another. If you want to find the vector from point \(A\) (position \(\mathbf{a}\)) to point \(B\) (position \(\mathbf{b}\)), you use the "End minus Start" rule:

\(\vec{AB} = \mathbf{b} - \mathbf{a}\)

Example: If \(A\) is at \((1, 2, 3)\) and \(B\) is at \((4, 0, 1)\):
\(\vec{AB} = \begin{pmatrix} 4 \\ 0 \\ 1 \end{pmatrix} - \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 3 \\ -2 \\ -2 \end{pmatrix}\)

Key Takeaway: \(\vec{AB} = -\vec{BA}\). Reversing the letters reverses the direction!

3. Vector Operations (The Algebra)

Vector Addition and Subtraction

To add or subtract vectors, simply add or subtract their corresponding components.

If \(\mathbf{u} = \begin{pmatrix} u_1 \\ u_2 \\ u_3 \end{pmatrix}\) and \(\mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}\), then \(\mathbf{u} + \mathbf{v} = \begin{pmatrix} u_1 + v_1 \\ u_2 + v_2 \\ u_3 + v_3 \end{pmatrix}\).

Geometrically: Adding vectors is like placing them "head-to-tail." If you follow vector \(\mathbf{u}\) and then follow vector \(\mathbf{v}\), the result \(\mathbf{u} + \mathbf{v}\) is the shortcut from where you started to where you ended.

Scalar Multiplication

When you multiply a vector by a "scalar" (a regular number \(k\)), you multiply every component by that number.

\(k \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} kx \\ ky \\ kz \end{pmatrix}\)

If \(k > 1\), the vector gets longer (stretches).
If \(0 < k < 1\), the vector gets shorter (shrinks).
If \(k\) is negative, the vector flips and points in the opposite direction.

The Zero Vector

The zero vector, denoted \(\mathbf{0}\), is \(\begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}\). It has no magnitude and no specific direction.

4. Magnitude and Unit Vectors

Magnitude (Length)

The magnitude of a vector \(\mathbf{v} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}\) is written as \(\lvert \mathbf{v} \rvert\). We use a 3D version of Pythagoras' Theorem to calculate it:

\(\lvert \mathbf{v} \rvert = \sqrt{x^2 + y^2 + z^2}\)

Unit Vectors

A unit vector is a vector that has a magnitude of exactly \(1\). We often denote it with a "hat," like \(\hat{\mathbf{v}}\).

To turn any vector into a unit vector, you divide the vector by its own magnitude:

\(\hat{\mathbf{v}} = \frac{\mathbf{v}}{\lvert \mathbf{v} \rvert}\)

Analogy: If you have a stick that is 5 meters long and you want to know which way it points but only need a 1-meter version of it, you would divide its length by 5.

5. Parallel Vectors

Two vectors are parallel if one is a scalar multiple of the other. That is, \(\mathbf{u}\) is parallel to \(\mathbf{v}\) if:

\(\mathbf{u} = k\mathbf{v}\) (for some constant \(k\))

If \(k\) is positive, they point in the same direction. If \(k\) is negative, they are still parallel but point in opposite directions.

Common Mistake: Students often forget that \(\begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}\) and \(\begin{pmatrix} -4 \\ 2 \\ -6 \end{pmatrix}\) are parallel because \(k = -2\). Always check if the components are in the same ratio!

6. Geometric Proofs using Vectors

One of the most powerful uses of vector algebra is proving geometric properties. To do this, we express the sides of a shape as vectors and use addition/subtraction to compare them.

Example: Proving a Parallelogram

To prove that a quadrilateral \(ABCD\) is a parallelogram, you only need to show that \(\vec{AB} = \vec{DC}\).

Why? If the vectors are equal, they must have the same magnitude (length) and the same direction (parallel), which is the definition of a parallelogram.

The Midpoint Formula

If point \(M\) is the midpoint of the line segment \(AB\), then its position vector \(\mathbf{m}\) is the average of the position vectors of \(A\) and \(B\):

\(\mathbf{m} = \frac{1}{2}(\mathbf{a} + \mathbf{b})\)

Quick Review: Key Takeaways

Vector vs Scalar: Vectors have magnitude and direction; scalars only have magnitude.
Standard Form: \(\mathbf{v} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}\) or \(\begin{pmatrix} x \\ y \\ z \end{pmatrix}\).
Displacement: \(\vec{AB} = \text{Position } B - \text{Position } A\).
Magnitude: \(\lvert \mathbf{v} \rvert = \sqrt{x^2 + y^2 + z^2}\).
Unit Vector: Divide a vector by its magnitude to get a length of 1.
Parallelism: \(\mathbf{u} = k\mathbf{v}\) means \(\mathbf{u} \parallel \mathbf{v}\).

Ready for more? In the next chapter, we will learn how to multiply vectors using the "Scalar Product" to find the angles between them!

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the fundamental difference between a scalar and a vector?

A scalar has only magnitude, whereas a vector has both magnitude and direction.

What are the two common ways to represent a vector in the IB DP curriculum?

The two ways are Column Vector Form, \(\mathbf{v} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}\), and Base Vector Form, \(\mathbf{v} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}\).

How is a position vector defined?

A position vector describes the location of a point relative to a fixed origin \(O(0,0,0)\), written as \(\vec{OP} = \mathbf{p}\).

What is the rule for finding the displacement vector \(\vec{AB}\) between two points?

The rule is "End minus Start": \(\vec{AB} = \mathbf{b} - \mathbf{a}\), where \(\mathbf{a}\) and \(\mathbf{b}\) are the position vectors of points \(A\) and \(B\).

How do you perform vector addition and subtraction algebraically?

You add or subtract the corresponding components of the vectors: \(\mathbf{u} \pm \mathbf{v} = \begin{pmatrix} u_1 \pm v_1 \\ u_2 \pm v_2 \\ u_3 \pm v_3 \end{pmatrix}\).

What happens to a vector when it is multiplied by a scalar \(k\)?

Every component of the vector is multiplied by \(k\). This changes the magnitude and, if \(k\) is negative, reverses the direction.

How do you calculate the magnitude (length) of a vector \(\mathbf{v}\)?

The magnitude is calculated using the 3D Pythagorean distance formula: \(|\mathbf{v}| = \sqrt{x^2 + y^2 + z^2}\).

How is a unit vector in the direction of \(\mathbf{v}\) calculated?

A unit vector \(\hat{\mathbf{v}}\) is found by dividing the vector by its magnitude: \(\hat{\mathbf{v}} = \frac{\mathbf{v}}{|\mathbf{v}|}\).

What is the mathematical condition for two vectors to be parallel?

Two vectors \(\mathbf{u}\) and \(\mathbf{v}\) are parallel if one is a scalar multiple of the other: \(\mathbf{u} = k\mathbf{v}\) for some constant \(k\).

How can vectors be used to prove a quadrilateral \(ABCD\) is a parallelogram?

Show that \(\vec{AB} = \vec{DC}\) (or \(\vec{AD} = \vec{BC}\)).

What is the vector midpoint formula?

The position vector \(\mathbf{m}\) of the midpoint of segment \(AB\) is \(\mathbf{m} = \frac{1}{2}(\mathbf{a} + \mathbf{b})\).

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