Scalar product formula: v · w = |v||w| cos θ
The scalar product formula is v · w = v₁w₁ + v₂w₂ + v₃w₃ = |v||w| cos θ.
What each letter means
- \(\mathbf v,\ \mathbf w\) two vectors
- \(\theta\) the angle between them
- \(|\mathbf v|,\ |\mathbf w|\) their magnitudes
When to use it
To find the angle between two vectors or two lines, and to test whether they are perpendicular (dot product 0).
Worked example
Find the angle between \(\mathbf v=\begin{pmatrix}1\\2\\2\end{pmatrix}\) and \(\mathbf w=\begin{pmatrix}2\\-1\\2\end{pmatrix}\).
- \(\mathbf v\cdot\mathbf w=2-2+4=4\); \(|\mathbf v|=|\mathbf w|=3\)
- \(\cos\theta=\dfrac{4}{9}\)
Answer: \(\theta=63.6^\circ\) (3 s.f.)
Common mistake
Dividing by only one magnitude when finding cos θ. Divide by |v| × |w|.
On your course
| Course | In the exam |
|---|---|
| AA HL | In the IB formula booklet |
| AI HL | In the IB formula booklet |
From our own IB Maths formula sheets, in our words. The IB gives its booklet to schools, so we checked against our own copy: your teacher has the official one. Official: IB DP Mathematics page.
Practise and revise
- Practise AA HL vector lines questions
- Practise AI HL scalar products and Voronoi questions
- Revise 3D vectors (AA HL)
- Revise 3D vectors (AI HL)
- Tool Vectors on your GDC
- Print AA HL one-page formula sheet
Questions
What is the scalar product formula?
The scalar product formula is v · w = v₁w₁ + v₂w₂ + v₃w₃ = |v||w| cos θ. v, w: two vectors; θ: the angle between them; | v|, | w|: their magnitudes.
Is the scalar (dot) product given in the exam?
AA HL: in the IB formula booklet. AI HL: in the IB formula booklet. This comes from our own IB Maths formula sheets; your teacher has the official booklet.
How can I tell if two vectors are perpendicular?
Their scalar product is 0, because cos 90° = 0.
Our own wording, examples and card, checked by IB Math Revision. Not produced or endorsed by the International Baccalaureate Organization.