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Vectors: A Level Maths knowledge organiser

Everything to know about vectors on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Vector
A quantity with size and direction, written as a column or as ai + bj + ck.
Magnitude
The length of a vector: √(a² + b² + c²).
Unit vector
A vector of length 1: divide a vector by its magnitude.
Position vector
The vector from the origin O to a point.

Key formulas

Magnitude; unit vector\(|x\mathbf i+y\mathbf j+z\mathbf k|=\sqrt{x^2+y^2+z^2},\) \(\hat{\mathbf a}=\frac{\mathbf a}{|\mathbf a|}\)
\(\overrightarrow{AB}\); distance \(AB\)\(\mathbf b-\mathbf a;\) \(|\mathbf b-\mathbf a|=\sqrt{\Delta x^2+\Delta y^2+\Delta z^2}\)

Worked example

A is (1, 2, 3) and B is (4, 6, 3). Find AB, its length and a unit vector in its direction.

  1. AB = b − a = 3i + 4j + 0k
  2. |AB| = √(9 + 16 + 0) = 5

Answer: AB = 3i + 4j, |AB| = 5, unit vector = 0.6i + 0.8j

Common mistakes

  • Sin θ = 2 not rejected
  • Interval not adjusted for multiple angles
  • Magnitude as a + b instead of √(a² + b²)
  • Distance in 3D missing the z-component

More on what examiners see students get wrong: Examiner Insights.

You should be able to…

  • Represent vectors as directed line segments, in column form and with i and j, and add, subtract and multiply vectors by scalars.
  • Use position vectors to find the vector between two points and the distance between them, and find the position vector of a midpoint.
  • Solve geometric problems with vectors, such as showing that three points are collinear or that two lines are parallel, by writing vectors in terms of others.
  • Use three-dimensional coordinates to plot points and find the distance between two points in 3D using an extension of Pythagoras' theorem.
  • Represent vectors in 3D with i, j and k, add and scale them, and find magnitudes, unit vectors and the angles a vector makes with the axes.
  • Use 3D vectors in mechanics contexts, such as finding the resultant of forces in three dimensions and the acceleration they produce.

The printable sheet

Vectors knowledge organiser for A Level Maths: one A4 page of key definitions, formulas, a worked example and common mistakes
Vectors knowledge organiser (A Level Maths), A4. Download the PDF.

Revise it next

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