Vectors, Lines & Planes (AA HL)
A vector carries both size and direction, which makes it the natural tool for describing points, directions and motion in two and three dimensions. This topic covers vector algebra and magnitude, the scalar and vector products and what each one is used for, and how to write and work with the vector and Cartesian equations of lines and planes - including deciding whether two lines meet, run parallel, or pass each other as skew lines.
What the syllabus says
This topic maps onto five points in the official IB Analysis & Approaches syllabus, all exclusive to HL.
| Code | Syllabus content |
|---|---|
| AHL3.12 | Concept of a vector; position and displacement vectors; representation using directed line segments; base vectors \(\mathbf i,\mathbf j,\mathbf k\). |
| AHL3.13 | The scalar (dot) product of two vectors and its properties; the angle between two vectors; perpendicular and parallel vectors. |
| AHL3.14 | Vector equation of a line in two and three dimensions: \(\mathbf r=\mathbf a+\lambda\mathbf b\), including the parametric and Cartesian forms. |
| AHL3.15 | Coincident, parallel, intersecting and skew lines, distinguishing between these cases; points of intersection. |
| AHL3.16 / AHL3.17 | The vector (cross) product of two vectors, and its use to find the area of a parallelogram or triangle. Vector and Cartesian equations of a plane. |
Intersections and angles between lines and planes (AHL3.18) build directly on this topic and often appear in the same exam question.
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 position vector?
A position vector describes where a point is relative to a fixed origin, written as \(\overrightarrow{OA}\) or just \(\mathbf a\). Its components are simply the coordinates of the point.
e.g. If \(A=(3,4,0)\), then \(\overrightarrow{OA}=(3,4,0)\) and \(|\overrightarrow{OA}|=\sqrt{9+16+0}=5\).
What is the scalar (dot) product?
The scalar product multiplies two vectors together to produce a single number: \(\mathbf a\cdot\mathbf b=a_1b_1+a_2b_2+a_3b_3\). It's the key to finding angles and testing perpendicularity.
e.g. \((1,2,2)\cdot(2,-1,2)=2-2+4=4\).
What is the vector (cross) product?
The vector product multiplies two vectors to produce a third vector, perpendicular to both of the originals. In three dimensions it's found with a determinant, and its magnitude gives an area.
e.g. \((1,0,0)\times(0,1,0)=(0,0,1)\), i.e. \(\mathbf i\times\mathbf j=\mathbf k\).
What does it mean for two vectors to be perpendicular?
Two non-zero vectors are perpendicular exactly when their scalar product is zero - this is the fastest way to check a right angle without measuring anything.
e.g. \((2,3)\cdot(3,-2)=6-6=0\), so \((2,3)\) and \((3,-2)\) are perpendicular.
What is a skew line?
Two lines in three dimensions are skew if they never meet and are not parallel - something that can't happen in two dimensions, where non-parallel lines always cross.
e.g. Direction vectors \((1,1,0)\) and \((2,2,0)\) are parallel (one is a scalar multiple of the other); \((1,0,0)\) and \((0,1,1)\) are not, so those two lines could be skew.
Key formulas
A handful of formulas cover almost every question on this topic. The two tables below summarise all of them at a glance - the explanations underneath go into more depth on each one.
Formula reference
The component forms of the scalar and vector products, and the line and plane equations, are printed in the official formula booklet.
| Formula | Used for | Booklet? |
|---|---|---|
| \(|\mathbf v|=\sqrt{x^2+y^2+z^2}\) | Magnitude of a vector | Not in booklet - prior knowledge |
| \(\mathbf a\cdot\mathbf b=a_1b_1+a_2b_2+a_3b_3\) | Scalar (dot) product, component form | ✓ Yes |
| \(\cos\theta=\dfrac{\mathbf a\cdot\mathbf b}{|\mathbf a||\mathbf b|}\) | Angle between two vectors | ✓ Yes |
| \(\mathbf a\times\mathbf b=|\mathbf a||\mathbf b|\sin\theta\,\hat{\mathbf n}\) | Vector (cross) product | ✓ Yes |
| Area \(=\tfrac12|\mathbf a\times\mathbf b|\) | Area of a triangle from two side vectors | ✓ Yes |
| \(\mathbf r=\mathbf a+\lambda\mathbf b\) | Vector equation of a line | ✓ Yes |
| \(ax+by+cz=d\) | Cartesian equation of a plane | ✓ Yes |
Scalar product vs vector product
Both combine two vectors, but they produce completely different kinds of answer and are used for different jobs.
| Feature | Scalar (dot) product | Vector (cross) product |
|---|---|---|
| Result | A scalar (a number) | A vector |
| Formula | \(\mathbf a\cdot\mathbf b=|\mathbf a||\mathbf b|\cos\theta\) | \(\mathbf a\times\mathbf b=|\mathbf a||\mathbf b|\sin\theta\,\hat{\mathbf n}\) |
| Used for | Angle between vectors; perpendicularity test (dot \(=0\)) | Area of a parallelogram/triangle; a normal vector to a plane |
| Defined in | 2D and 3D | 3D only |
Vector algebra
Before combining vectors, you need to be fluent with the basic operations on their components.
Magnitude
\[|\mathbf v|=\sqrt{v_1^2+v_2^2+v_3^2}\]
The length of a vector - always a non-negative number, found with Pythagoras extended to three dimensions.
Unit vector
\[\hat{\mathbf v}=\dfrac{1}{|\mathbf v|}\mathbf v\]
A vector of length 1 pointing in the same direction as \(\mathbf v\) - divide every component by the magnitude.
Addition and scalar multiplication
Add or subtract component by component; multiplying by a scalar \(k\) scales every component by \(k\), stretching or reversing the vector.
Products of vectors
Two different ways of "multiplying" vectors, each with its own geometric meaning.
Scalar product and angle
\[\cos\theta=\dfrac{\mathbf a\cdot\mathbf b}{|\mathbf a||\mathbf b|}\]
Rearrange the dot product formula to isolate \(\theta\), then take the inverse cosine.
Perpendicularity test
\(\mathbf a\cdot\mathbf b=0\) if and only if \(\mathbf a\) and \(\mathbf b\) are perpendicular (assuming neither is the zero vector) - no angle calculation needed.
Cross product and area
The magnitude \(|\mathbf a\times\mathbf b|\) is the area of the parallelogram spanned by \(\mathbf a\) and \(\mathbf b\); halve it for the area of the triangle they form.
Lines and planes
Once you can add, scale and multiply vectors, you can describe entire lines and planes rather than single points.
Vector equation of a line
\[\mathbf r=\mathbf a+\lambda\mathbf b\]
\(\mathbf a\) is a position vector of any point on the line; \(\mathbf b\) is a direction vector; \(\lambda\) ranges over all real numbers.
Intersecting, parallel or skew
Two lines are parallel if their direction vectors are scalar multiples; otherwise set the two equations equal component-wise - a consistent solution means they intersect, no solution means they're skew.
Cartesian equation of a plane
\[ax+by+cz=d\]
\((a,b,c)\) is a normal vector to the plane; find \(d\) by substituting any known point on the plane.
Worked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
Find \(|\mathbf v|\) and a unit vector in the direction of \(\mathbf v=(4,-4,2).\)
Worked solution
\(|\mathbf v|=\sqrt{16+16+4}=\sqrt{36}\) M1
\(|\mathbf v|=6.\) A1
Unit vector \(\hat{\mathbf v}=\tfrac16(4,-4,2)=(\tfrac23,-\tfrac23,\tfrac13).\) A1
Find \((2,1,-1)\times(1,0,3).\)
Worked solution
(a) \(\begin{vmatrix}\mathbf i&\mathbf j&\mathbf k\\2&1&-1\\1&0&3\end{vmatrix}.\) M1
\(\mathbf i(1\cdot3-(-1)\cdot0)-\mathbf j(2\cdot3-(-1)\cdot1)+\mathbf k(2\cdot0-1\cdot1).\) A1
\(=(3,-7,-1).\) 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.
- Writing the cross product as a number. The vector product always produces a vector - give the answer as \((x,y,z)\) or in \(\mathbf i,\mathbf j,\mathbf k\) form, never as a single scalar.
- Forgetting to take the inverse cosine. The dot product formula gives \(\cos\theta\) - the final answer needs \(\theta=\cos^{-1}(\ldots)\), not the cosine value itself.
- Assuming non-intersecting lines must be parallel. In three dimensions, two lines can fail to meet without being parallel - that's exactly what a skew line is.
- Sign errors in the cross-product determinant. The middle (\(\mathbf j\)) component of the expansion is negated - it's easy to forget the minus sign there.
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/cross products and the angle between vectors, without coordinate algebra by hand.
- Enter each vector's components (as a list or a 3×1 matrix).
- Store components in lists; magnitude = √(sum of squares); dot product = sum(L1×L2).TI-84
- Type vectors with the matrix template; use dotP(), crossP() and norm() from menu → Matrix & Vector → Vector.Nspire
- Run-Matrix → 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, lines & planes questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
What's the difference between the scalar product and the vector product?
The scalar (dot) product of two vectors gives a number, and is used to find the angle between vectors or to test perpendicularity (dot product zero). The vector (cross) product gives another vector, perpendicular to both originals, and is used to find a normal to a plane or the area of a parallelogram or triangle.
How do I tell if two lines in 3D are skew?
First check the direction vectors: if they're not scalar multiples of each other, the lines aren't parallel. Then try to solve the system from setting the two vector equations equal - if there's no solution that satisfies all three component equations, the lines don't meet, so they're skew.
Is the vector product examined on Paper 1, without a calculator?
Yes - the determinant method for the cross product is pure algebra, so it appears on both papers. Paper 2 questions tend to combine it with numerical follow-up work like finding an area or an angle.
How do I find the equation of a plane through three points?
Form two vectors that lie in the plane (e.g. \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\) from the three points), take their cross product to get a normal vector \(\mathbf n\), then substitute one of the points into \(\mathbf r\cdot\mathbf n=\mathbf a\cdot\mathbf n\) to get the Cartesian equation. See the GDC guide for how to compute the cross product directly.
Sub-topics
Vectors, Lines & Planes broken down into its individual skills, each with its own focused page.
Related topics
More Geometry & Trigonometry topics from the same AA HL syllabus unit, in case you want to keep going.