Vector fundamentals and dot product

IB Mathematics: Analysis and Approaches HL· IB AA HL Syllabus 3.12· 15 min read

1. Vector Fundamentals in 2D and 3D★★☆☆☆⏱ 4 min

📘 Definition

Position Vector

\vec{v} = \begin{pmatrix} v_x \\ v_y \\ v_z \end{pmatrix} = v_x \hat{i} + v_y \hat{j} + v_z \hat{k}

A position vector connects the origin to a point $(v_{x}, v_{y}, v_{z})$ in 3D space, with components equal to the coordinates of the point.

Example:

The position vector of point $(2, - 3, 1)$ is $2 \hat{i} - 3 \hat{j} + \hat{k}$ .

The magnitude (length) of a vector is calculated using the 3D extension of Pythagoras' theorem:

|\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}

A unit vector in the direction of $oxed{\vec{v}}$ is found by dividing $oxed{\vec{v}}$ by its own magnitude.

📐 Worked Example

Find the unit vector in the direction of $v = 2 \hat{i} - \hat{j} + 4 \hat{k}$

1
First calculate the magnitude of $v$ :
2
$|\vec{v}| = \sqrt{2^2 + (-1)^2 + 4^2} = \sqrt{4 + 1 + 16} = \sqrt{21}$
3
Divide each component of $v$ by the magnitude to get the unit vector:
4
$\hat{v} = \frac{1}{\sqrt{21}} (2\hat{i} - \hat{j} + 4\hat{k}) = \frac{2}{\sqrt{21}}\hat{i} - \frac{1}{\sqrt{21}}\hat{j} + \frac{4}{\sqrt{21}}\hat{k}$

2. Algebraic Definition of the Dot Product★★☆☆☆⏱ 5 min

📘 Definition

Dot Product (Scalar Product)

\vec{v} \cdot \vec{w}

For two vectors $v = (v_{x}, v_{y}, v_{z})$ and $w = (w_{x}, w_{y}, w_{z})$ , the dot product is the sum of the products of corresponding components.

Example:

If $v = (1, 2)$ and $w = (3, 4)$ , then $v \cdot w = (1) (3) + (2) (4) = 11$ .

\vec{v} \cdot \vec{w} = v_x w_x + v_y w_y + v_z w_z

The dot product follows standard algebraic properties: it is commutative ( $v \cdot w = w \cdot v$ ) and distributive over vector addition ( $v \cdot (w + u) = v \cdot w + v \cdot u$ ).

📐 Worked Example

Calculate $a \cdot b$ where $a = 3 \hat{i} - 2 \hat{k}$ and $b = - \hat{i} + 4 \hat{j} + \hat{k}$

1
Write out all components, including the zero $y$ -component for $a$ : $a = (3, 0, - 2)$ , $b = (- 1, 4, 1)$
2
Multiply corresponding components and sum the results:
3
$\vec{a} \cdot \vec{b} = (3)(-1) + (0)(4) + (-2)(1) = -3 + 0 - 2 = -5$

3. Geometric Interpretation: Angles Between Vectors★★★☆☆⏱ 6 min

The geometric definition of the dot product connects it to the angle $θ$ between two vectors placed tail-to-tail, where $0 \leq θ \leq 18 0^{\circ}$ :

\vec{v} \cdot \vec{w} = |\vec{v}||\vec{w}|\cos\theta

Rearranging this formula gives us a way to calculate the angle between any two non-zero vectors:

\cos\theta = \frac{\vec{v} \cdot \vec{w}}{|\vec{v}||\vec{w}|}

📐 Worked Example

Find the angle between $p = (2, 1, - 1)$ and $q = (1, - 1, 1)$ , to the nearest degree

1
Calculate the dot product of the two vectors:
2
$\vec{p} \cdot \vec{q} = (2)(1) + (1)(-1) + (-1)(1) = 2 - 1 - 1 = 0$
3
Since the dot product is zero and both vectors are non-zero, they are perpendicular:
4
$\theta = 90^\circ$

✓ Quick check

Test your understanding:

What is the angle between two non-zero vectors with $v \cdot w = - ∣ v ∣∣ w ∣$ ?
- 0°
- 90°
- 180°
- 45°
Reveal answer
180° —
Correct! Rearranging gives $cos θ = - 1$ , so $θ = 18 0^{\circ}$ , meaning the vectors point in opposite directions.

4. Scalar Projection of Vectors★★★☆☆⏱ 5 min

📘 Definition

Scalar Projection

The scalar projection of $v$ onto $w$ is the signed length of the component of $v$ that lies along the direction of $w$ .

\text{proj}_{\vec{w}} \vec{v} = \frac{\vec{v} \cdot \vec{w}}{|\vec{w}|}

A positive projection means the component points in the same direction as $w$ , while a negative projection means it points in the opposite direction.

📐 Worked Example

Calculate the scalar projection of $v = (3, 4, 0)$ onto $w = (1, 1, 0)$

1
First calculate the dot product and magnitude of $w$ :
2
$\vec{v} \cdot \vec{w} = (3)(1) + (4)(1) + 0 = 7, \quad |\vec{w}| = \sqrt{1^2 + 1^2} = \sqrt{2}$
3
Substitute into the projection formula and rationalize:
4
$\text{proj}_{\vec{w}} \vec{v} = \frac{7}{\sqrt{2}} = \frac{7\sqrt{2}}{2}$

5. Common Pitfalls

Wrong move:

Forgetting to include zero components when calculating the dot product

Why:

Vectors written only with non-zero components still have zero components for missing axes that contribute to the calculation

Correct move:

Always write out all components (including zeros) for 3D vectors before calculating the dot product

Wrong move:

Treating the dot product result as a vector instead of a scalar

Why:

The dot product (scalar product) always outputs a single number, not a multi-component vector

Correct move:

Remember the name: scalar product outputs a scalar, cross product outputs a vector

Wrong move:

Using the wrong calculator angle mode for angle questions

Why:

Calculators output different results for radians vs degrees, leading to incorrect final answers

Correct move:

Always check your calculator's angle mode matches the requirement stated in the question

Wrong move:

Assuming a zero dot product means one vector is the zero vector

Why:

The most common case of a zero dot product is two non-zero perpendicular vectors

Correct move:

If the dot product is zero and both vectors are non-zero, conclude they are perpendicular

6. Quick Reference Cheatsheet

Concept	Formula
Vector magnitude	$∣ v ∣ = v_{x}^{2} + v_{y}^{2} + v_{z}^{2}$
Unit vector	$\overset{v}{^} = \frac{v}{∣ v ∣}$
Dot product (algebraic)	$v \cdot w = v_{x} w_{x} + v_{y} w_{y} + v_{z} w_{z}$
Dot product (geometric)	$v \cdot w = ∣ v ∣∣ w ∣ cos θ$
Angle between vectors	$cos θ = \frac{v \cdot w}{∣ v ∣∣ w ∣}$
Perpendicular condition	$v \cdot w = 0 (v, w \neq = 0)$
Scalar projection	$proj_{w} v = \frac{v \cdot w}{∣ w ∣}$

7. Frequently Asked

Is the dot product of two vectors a vector or a scalar?

The dot product (also called the scalar product) always produces a scalar value, unlike the cross product which produces a vector. This is a key distinction you must remember for exams.

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · 1
Find angle between two 3D vectors
2023 · 2
Prove two lines are perpendicular
2024 · 1
Calculate scalar projection of vector

Going deeper

Official SyllabusIB AA HL Geometry & TrigonometryOfficial curriculum reference

What's Next

Vector fundamentals and the dot product form the foundation for all further vector topics in IB AA HL. You will use the dot product constantly when working with lines and planes in 3D space, from finding angles between lines to calculating distances from points to planes. These skills also underpin cross product calculations, which are used for finding areas of parallelograms and normals to planes. Mastery of these core concepts will make more advanced vector geometry problems much more manageable, as almost all vector problems rely on correctly calculating dot products and interpreting their results.