Geometry

Vectors

Different journeys. The same displacement.

Predict, test and explain

Step 1 of 3

Predict

Add vectors (3, 1) and (−1, 3). What is their resultant?

1a+b=(24)1\mathbf a+\mathbf b=\binom{2}{4}
−10-10
−10-10
−8-8
−8-8
−6-6
−6-6
−4-4
−4-4
−2-2
−2-2
22
22
44
44
66
66
88
88
1010
1010
xx
yy

Scroll the diagram sideways to see every label.

a=(31)\mathbf a=\binom{3}{1}b=(−13)\mathbf b=\binom{-1}{3}

Place the second vector at the end of the first. The green arrow is the resultant.

InvestigateSwap the order of the journeys. What stays the same?

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.

Add the two displacement vectors.

a=(2−1),b=(−34)\mathbf a=\binom{2}{-1},\qquad\mathbf b=\binom{-3}{4}

Hint

Add the horizontal components, then add the vertical components.

Reveal answer

The combined displacement is one unit left and three units up.

a+b=(2−3−1+4)=(−13)\mathbf a+\mathbf b=\binom{2-3}{-1+4}=\binom{-1}{3}

Build an explanation

Hint 1

Read each vector as an across movement and an up movement.

Hint 2

Place the second journey at the end of the first.

Hint 3

Add corresponding components; multiply both components when scaling a vector.

Worked example

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

Add a = (3, 1) and b = (−1, 3), written as across/up pairs.

  1. Add across movements: 3 + (−1) = 2.
  2. Add up movements: 1 + 3 = 4.

Answer: a + b = (2, 4).

Connect this idea

Watch out: A vector describes displacement, so moving its starting point does not change its components.