A-level · Beta

Vectors and combinations

Drag vectors and explore scaling, addition, angles and collinearity.

Drag U and V to change their components. The green arrow is λu + v.

λu+v=(13)\lambda\mathbf{u}+\mathbf{v}=\begin{pmatrix}1\\3\end{pmatrix}
Drag the labelled rings or use their arrow keys.-6-4-2246-6-4-2246xy0UV

Tab to a handle, then use the arrow keys. Use the sliders for precise values.

The vectors are not collinear.

u and v
(21),(−12)\begin{pmatrix}2\\1\end{pmatrix},\quad \begin{pmatrix}-1\\2\end{pmatrix}
Dot product
u⋅v=0\mathbf{u}\cdot\mathbf{v}=0
Angle
≈90∘\approx 90^\circ

The red arrow is also drawn from the tip of λu, showing addition head to tail. Components and dot products are exact; displayed angles are approximate.

Predict, test and explain

Step 1 of 3

Predict

For u = (2, 1) and v = (−1, 2), what is u · v?

InvestigateCan you make u and v collinear?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Try it in the simulation. Change one thing at a time.

Notes and saved values stay in this tab. They disappear when you reload.

Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

For u = (2, 1) and v = (−1, 2), what is u · v?

(21)⋅(−12)\begin{pmatrix}2\\1\end{pmatrix}\cdot\begin{pmatrix}-1\\2\end{pmatrix}

Hint

Add corresponding components to add vectors.

Reveal answer

2 × (−1) + 1 × 2 = 0, so these nonzero vectors are perpendicular.

00

Build an explanation

Hint 1

Add corresponding components to add vectors.

Hint 2

A scalar multiplies every component.

Hint 3

A zero dot product gives a right angle when both vectors are nonzero.

Worked example

This example uses fixed values, separate from the diagram controls.

Find 2u + v for u = (2, 1), v = (−1, 2).

  1. Double u to get (4, 2).
  2. Add v component by component.
  3. The result is (4 − 1, 2 + 2).

Answer: (3, 4).

Connect this idea

Watch out: The dot product is a scalar sum of component products; zero vectors have no defined angle.

Specification and learning route

AQA J2, J3, J4 · Edexcel Pure 10.2, Pure 10.3, Pure 10.4

AS core; A-level extension. This model illustrates selected content; references are not a claim of full coverage or exam-board endorsement.

Useful starting knowledge: Coordinates; Pythagoras; Trigonometry; Scalar arithmetic.

AQA specification · Edexcel specification