Dot Product and Angles (AI HL)
The scalar (dot) product turns two vectors into a single number that encodes how aligned they are. It's the fastest route to the angle between two vectors, and a one-line test for perpendicularity. This page covers the formula, when to use each version of it, and the mistakes that cost marks. It's part of the broader Vectors topic.
12 questions on this sub-topic.
The scalar product formula
Covered under IB syllabus reference AHL3.13: definition and calculation of the scalar product, and its use to find the angle between two vectors and to test for perpendicularity.
Scalar product / angle between vectors
\[\mathbf v\cdot\mathbf w=|\mathbf v||\mathbf w|\cos\theta\]
In components, \(\mathbf v\cdot\mathbf w\) is found by multiplying matching entries and adding the results. Rearrange for \(\cos\theta\) to find the angle between two vectors.
✓ In the formula bookletPerpendicularity test
Two non-zero vectors \(\mathbf v\) and \(\mathbf w\) are perpendicular exactly when \(\mathbf v\cdot\mathbf w = 0\). This follows straight from the formula above, since \(\cos 90^\circ = 0\) - no need to find either magnitude first.
Need the full syllabus wording, the cross product, or GDC angle-mode tips? See Vectors.
Worked examples
Find \((2,1,-1)\cdot(3,0,4).\)
Worked solution
\((2,1,-1)\cdot(3,0,4) = 2\cdot3 + 1\cdot0 + (-1)\cdot4\) M1
\(= 2.\) A1
Find the angle between \(\mathbf a = (1,2,2)\) and \(\mathbf b = (2,0,1)\).
Worked solution
\(\mathbf a\cdot\mathbf b = 2 + 0 + 2\) M1 \(= 4.\) A1
\(\cos\theta = \dfrac{\mathbf a\cdot\mathbf b}{|\mathbf a||\mathbf b|} = \dfrac{4}{3\sqrt5}.\) M1 \(\theta \approx 53.4^\circ.\) A1
Find the vector projection of \(\mathbf a=(3,4,0)\) onto \(\mathbf b=(0,1,0).\)
Worked solution
\(\text{proj}_{\mathbf b}\mathbf a = \dfrac{\mathbf a\cdot\mathbf b}{|\mathbf b|^2}\mathbf b.\) M1
\(\mathbf a\cdot\mathbf b = 4.\) A1
(\(|\mathbf b|^2 = 1\)): M1
\(\dfrac{4}{1}(0,1,0)\) A1
\(= (0,4,0).\) A1
Common mistakes
- Treating the dot product as a vector. \(\mathbf v\cdot\mathbf w\) is always a single number (a scalar) - it's the cross product, not the dot product, that produces another vector.
- Overcomplicating the perpendicularity test. There's no need to find \(|\mathbf v|\), \(|\mathbf w|\), or the angle itself to show two vectors are perpendicular - just show \(\mathbf v\cdot\mathbf w = 0\) directly from the components.
- Wrong angle mode on the GDC. \(\cos^{-1}\) of a component ratio gives an answer in whatever mode the calculator is set to - check degrees vs radians before evaluating, or a fully correct method still scores the wrong final value.
Ready to practise properly?
12 dot-product and angle questions, marked instantly like the real exam.
Quick answers
What is the formula for the scalar product of two vectors?
\(\mathbf v\cdot\mathbf w=|\mathbf v||\mathbf w|\cos\theta\), where \(\theta\) is the angle between the two vectors. In components, \(\mathbf v\cdot\mathbf w\) is found by multiplying matching components and adding the results.
How do you test whether two vectors are perpendicular?
Two non-zero vectors are perpendicular exactly when their scalar product is zero: \(\mathbf v\cdot\mathbf w = 0\). This works directly from the components, with no need to find magnitudes or angles first.