Vector magnitude formula: |v| = √(v₁² + v₂² + v₃²)
The vector magnitude formula is |v| = √(v₁² + v₂² + v₃²).
What each letter means
- \(\mathbf v\) the vector, with components v₁, v₂, v₃
- \(|\mathbf v|\) its magnitude (length)
When to use it
To find the length of a vector, a distance between position vectors, a speed from a velocity vector, or a unit vector v/|v|.
Worked example
Find \(|\mathbf v|\) and a unit vector in the direction of \(\mathbf v=\begin{pmatrix}2\\-3\\6\end{pmatrix}\).
- \(|\mathbf v|=\sqrt{4+9+36}=\sqrt{49}=7\)
- Unit vector \(=\tfrac17\mathbf v\)
Answer: \(|\mathbf v|=7\); unit vector \(\begin{pmatrix}2/7\\-3/7\\6/7\end{pmatrix}\)
Common mistake
Adding the components before squaring. Square each component, add, then square root.
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 vectors and kinematics 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 vector magnitude formula?
The vector magnitude formula is |v| = √(v₁² + v₂² + v₃²). v: the vector, with components v₁, v₂, v₃; | v|: its magnitude (length).
Is the magnitude of a vector 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 do I find a unit vector?
Divide the vector by its magnitude: v / |v|. The result has length 1 and points the same way.
Our own wording, examples and card, checked by IB Math Revision. Not produced or endorsed by the International Baccalaureate Organization.