A Level Maths: Vectors in 3-d Topic Summary and Resources
Video Lessons
Watch alongside the worksheet for the full lesson experience, then test your understanding with the lesson questions.
Revision Notes
Handwritten notes summarising the key ideas for each lesson. Ideal for quick review before a test.
Exam Questions
Past-paper-style questions organised by topic, with full mark schemes.
Drawn from OCR and Edexcel past papers but designed to be useful for students of all UK exam boards — including AQA and OCR MEI — unless a sheet is explicitly board-specific.
A Level Maths vectors in 3-d extends Vectors to three dimensions, introducing the unit vector $\mathbf{k}$ alongside $\mathbf{i}$ and $\mathbf{j}$. The conceptual leap is small but the applications are huge — 3D vectors underpin physics, engineering, computer graphics, and Vector Mechanics.
You write 3D vectors as column vectors or as $a\mathbf{i} + b\mathbf{j} + c\mathbf{k}$. You find magnitudes using the extended Pythagorean formula $\sqrt{a^2 + b^2 + c^2}$, and distances between points in 3D using the distance formula $d^2 = (x_1-x_2)^2 + (y_1-y_2)^2 + (z_1-z_2)^2$. You add, subtract, and scale 3D vectors exactly as in 2D, and identify parallel vectors as scalar multiples of each other. You use 3D vectors to solve geometric problems — finding position vectors of points, identifying shapes formed by points in 3D space, and showing that points are collinear or that lines are parallel.
Vectors in 3-d is part of the Pure Maths strand of A Level Maths for AQA, Edexcel, OCR, and OCR MEI students.
Watch out for…
A few things to be careful with: the magnitude in 3D includes ALL three components squared and summed under the square root — a common slip is leaving one out; for a unit vector in 3D, divide by the 3D magnitude, not the 2D one; vectors that look 'parallel-ish' geometrically may not actually be scalar multiples — check all three ratios match; and when calculating the distance between two 3D points, square the DIFFERENCE of each coordinate, not the coordinates themselves.