Vectors (AI HL)
A vector carries both a size and a direction, which makes it the natural way to describe position, velocity, and force. This topic covers vector components and magnitude, unit vectors, the scalar (dot) product for finding angles and testing perpendicularity, and using vectors to model motion - position, velocity, relative velocity and constant or variable acceleration.
What the syllabus says
This topic maps onto three points in the official IB Applications & Interpretation syllabus, examined only at HL.
| Code | Syllabus content |
|---|---|
| AHL3.10 | Concept of a vector and a scalar. Representation of vectors using directed line segments. Sum and difference of two vectors, multiplication by a scalar, magnitude of a vector \(|\mathbf v|\) from components, unit vectors, base vectors \(\mathbf i,\mathbf j,\mathbf k\), and rescaling/normalizing vectors. |
| AHL3.12 | Vector applications to kinematics: modelling linear motion with constant velocity, \(\mathbf r=\mathbf r_0+\mathbf v t\); relative position \(\overrightarrow{AB}\); motion with variable velocity in two dimensions. |
| AHL3.13 | Definition and calculation of the scalar product \(\mathbf v\cdot\mathbf w=|\mathbf v||\mathbf w|\cos\theta\), used to find the angle between two vectors and to test for perpendicularity (\(\mathbf v\cdot\mathbf w=0\)). Definition and calculation of the vector (cross) product \(\mathbf v\times\mathbf w=|\mathbf v||\mathbf w|\sin\theta\,\hat{\mathbf n}\), and its use to find the area of a parallelogram or triangle via \(|\mathbf v\times\mathbf w|\). |
The vector equation of a line (AHL3.11) and the vector product (also AHL3.13) build directly on the content here.
Key terms
Five words worth knowing cold before you touch the formulas below - each with a worked example showing exactly what it means.
What is a vector?
A vector is a quantity with both magnitude and direction, usually written in component form. It's represented by a directed line segment - the length shows the magnitude, the arrow shows the direction.
e.g. A displacement of \((3,4)\) means 3 units across and 4 units up from the start point.
What is the magnitude of a vector?
The magnitude, \(|\mathbf v|\), is the vector's length - found using Pythagoras' theorem on its components: \(|\mathbf v|=\sqrt{x^2+y^2+z^2}\).
e.g. For \(\mathbf v=(2,6,9)\), \(|\mathbf v|=\sqrt{4+36+81}=\sqrt{121}=11\).
What is the scalar (dot) product?
The scalar product of two vectors, \(\mathbf a\cdot\mathbf b = a_1b_1+a_2b_2+a_3b_3\), combines them into a single number - it's used to find the angle between vectors and to test perpendicularity.
e.g. For \(\mathbf a=(3,-2,5)\), \(\mathbf b=(1,4,2)\): \(\mathbf a\cdot\mathbf b = 3-8+10=5\).
What is a unit vector?
A unit vector has magnitude exactly \(1\) and points in the same direction as the original vector. Find it by dividing every component by the vector's magnitude.
e.g. For \(\mathbf v=(5,-12)\), \(|\mathbf v|=13\), so the unit vector is \((5/13,-12/13)\approx(0.385,-0.923)\).
What is vector kinematics?
Vector kinematics models the position of a moving object over time. With constant velocity, \(\mathbf r(t) = \mathbf r_0+\mathbf v t\); differentiating position gives velocity, and integrating velocity gives position.
e.g. For \(\mathbf r(t)=(2+3t,\,1+4t)\), \(\mathbf r(5)=(2+15,\,1+20)=(17,21)\).
Key formulas
Four core formulas cover magnitude, direction and motion - plus one derived formula for relative velocity.
Formula reference
Magnitude, unit vectors, constant-velocity kinematics and the scalar product are all on the AI formula booklet.
| Formula | Used for | Booklet? |
|---|---|---|
| \(|\mathbf v| = \sqrt{x^2+y^2+z^2}\) | Magnitude of a vector | ā Yes |
| \(\hat{\mathbf v} = \dfrac{\mathbf v}{|\mathbf v|}\) | Unit vector | ā Yes |
| \(\mathbf r(t)=\mathbf r_0+\mathbf v t\) | Position with constant velocity | ā Yes |
| \(\mathbf v\cdot\mathbf w=|\mathbf v||\mathbf w|\cos\theta\) | Scalar product / angle between vectors | ā Yes |
| \(\mathbf v_{B/A}=\mathbf v_B-\mathbf v_A\) | Velocity of B relative to A | Not in the formula booklet - derived from vector subtraction |
Vectors vs scalars
Whether a quantity has direction attached is the single question that decides which type it is.
| Feature | Scalar | Vector |
|---|---|---|
| Has direction? | No | Yes |
| Example | Speed (\(5\) m/s) | Velocity (\((3,4)\) m/s) |
| Combining two of them | Ordinary addition | Component-wise addition (or the triangle law) |
| Comparison of size | Direct | Compare magnitudes, \(|\mathbf v|\) |
Working with vectors
Before combining or comparing vectors, you need to be fluent with these three basic operations.
Magnitude
\[|\mathbf v| = \sqrt{x^2+y^2+z^2}\]
Pythagoras' theorem extended to any number of dimensions - always non-negative.
ā In the formula bookletUnit vector
\[\hat{\mathbf v} = \dfrac{1}{|\mathbf v|}\mathbf v\]
Divide every component by the magnitude to rescale a vector to length \(1\) without changing its direction.
ā In the formula bookletAddition and scalar multiples
\[k\mathbf a + \mathbf b\]
Multiply each component by \(k\), then add or subtract component-wise.
Not in the formula booklet - definitionVectors in kinematics
Position, velocity and relative motion are all handled the same way once you treat displacement as a vector.
Constant velocity
\[\mathbf r(t) = \mathbf r_0 + \mathbf v t\]
Position at time \(t\) equals the starting position plus velocity times time.
ā In the formula bookletRelative velocity
\[\mathbf v_{B/A} = \mathbf v_B - \mathbf v_A\]
The velocity of B as seen from A - subtract A's velocity from B's.
Not in the formula booklet - derivedVariable velocity
\[\mathbf r(t) = \mathbf r(0) + \int_0^t \mathbf v(t)\,dt\]
When velocity isn't constant, integrate each component of \(\mathbf v(t)\) to recover position.
Not in the formula booklet - calculus applied to vectorsThe scalar product
The scalar product is the workhorse for anything involving the angle between two vectors.
Angle between vectors
\[\cos\theta = \dfrac{\mathbf v\cdot\mathbf w}{|\mathbf v||\mathbf w|}\]
Rearrange the scalar product formula and take the inverse cosine.
ā In the formula bookletTesting perpendicularity
\[\mathbf v\cdot\mathbf w = 0 \iff \mathbf v \perp \mathbf w\]
A dot product of zero, with both vectors non-zero, means the angle between them is exactly \(90^\circ\).
ā Direct consequence of the scalar product formulaWorked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
\(\mathbf v=\begin{pmatrix}2\\6\\9\end{pmatrix}.\)
(a) Find its magnitude (3 s.f.).
(b) Find the unit vector in the direction of \(\mathbf v.\)
Worked solution
(a) \(|\mathbf v|=\sqrt{2^2+6^2+9^2}\) M1
\(=\sqrt{121}=11.\) A1
(b) \((2/11,6/11,9/11).\) A1
A particle has position \(\mathbf r(t) = (2 + 3t,\ 1 + 4t)\) m, \(t\) in seconds.
(a) Find the velocity vector.
(b) Find the speed.
(c) Find the position after 5 s.
Worked solution
(a) \(\mathbf v = \dfrac{d\mathbf r}{dt}\) M1
\(= (3, 4)\) m/s. A1
(b) Speed \(= \sqrt{9 + 16}\) M1
\(= 5\) m/s. A1
(c) \(\mathbf r(5) = (17, 21)\) m. M1 A1
Common mistakes
The four slip-ups that account for most of the marks lost on this topic - worth reading before you start practising, not just after you get one wrong.
- Treating magnitude as a vector. \(|\mathbf v|\) is a single number (a scalar) - don't write it in component form or try to add it to another vector.
- Forgetting to subtract in the right order for relative velocity. The velocity of B relative to A is \(\mathbf v_B - \mathbf v_A\), not \(\mathbf v_A - \mathbf v_B\) - the order matters and reverses the direction.
- Assuming two objects collide just because their paths cross. Paths crossing only means the positions are equal at possibly different times - a genuine collision needs the same \(t\) to satisfy both the \(x\)- and \(y\)-equations simultaneously.
- Using the constant-velocity formula for variable velocity. \(\mathbf r(t) = \mathbf r_0+\mathbf v t\) only holds when velocity is constant - for variable velocity, you need to integrate \(\mathbf v(t)\) component by component instead.
Using your GDC
Every step below is a real button sequence, not a vague "use your calculator" hint - covering the TI-84 Plus, TI-Nspire, and Casio fx-9860/fx-CG50. Pick your model to filter down to just the steps that apply to you.
Lengths, dot products and the angle between vectors without coordinate algebra - useful on every vector question once you've entered the components.
- Enter each vector's components (as a list or a 3Ć1 matrix).
- Store components in lists; magnitude = ā(sum of squares); dot product = sum(L1ĆL2).TI-84
- Type vectors with the matrix template; use dotP(), crossP() and norm() from menu ā Matrix & Vector ā Vector.Nspire
- Run-Matrix ā MAT/VCT to enter vectors; OPTN gives DotP, CrossP and the norm (magnitude).Casio
- Angle between: \(\cos\theta = (\mathbf a\cdot\mathbf b)/(|\mathbf a||\mathbf b|)\), then inverse cosine.
Tip: A dot product of 0 means the vectors are perpendicular - the quickest perpendicularity check.
See the full GDC guide for more calculator models and topics.
Ready to practise properly?
Vectors questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
What's the difference between a vector and a scalar?
A scalar is a single number with magnitude only, like a temperature or a speed. A vector has both magnitude and direction, like a velocity or a force - it's usually written in component form, e.g. \((3,4)\).
How do I find the magnitude of a vector?
Square each component, add them, then take the square root: \(|\mathbf v| = \sqrt{x^2+y^2+z^2}\) for a 3D vector (drop the \(z^2\) term in two dimensions).
What does a dot product of zero tell me?
It tells you the two vectors are perpendicular. Since \(\mathbf v\cdot\mathbf w = |\mathbf v||\mathbf w|\cos\theta\), a zero dot product with non-zero vectors forces \(\cos\theta=0\), so \(\theta=90^\circ\).
How does vector kinematics work?
A moving object's position is modelled as \(\mathbf r(t) = \mathbf r_0 + \mathbf v t\) for constant velocity. Differentiate position to get velocity, or integrate velocity to get position, for objects moving with variable velocity.
Sub-topics
Vectors broken down into its individual skills, each with its own focused page.
Related topics
More Geometry & Trigonometry topics from the same AI HL syllabus unit, in case you want to keep going.