Math With AmarA C A D E M Y
All math animations

High school · Vectors

Add vectors head to tail

Slide a displacement without turning it, then connect the starting point to the final destination.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Add vectors head to tail. Final sum a + b: (3, 3). Length of the sum: 4.243. Current tail of b: (0, 0)Same vector. New starting point.ab0xAmber: final sum · Teal: translating b
These are displacement vectors in a Euclidean plane with the same unit on both axes. Translating an arrow preserves the vector. The amber resultant is the final sum; only the teal arrow's placement changes during playback.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Final sum a + b
(3, 3)
Length of the sum
4.243
Current tail of b
(0, 0)

HD animation studio

From experiment to screen.

Present a crisp Canvas scene, save a full-HD image, or capture your model as a silent video.

Understand what you are seeing

The idea behind the motion.

A vector records a change in position, not an address. Moving its tail does not change its horizontal and vertical components. Place the second arrow at the first arrow's head to see why adding components gives the overall displacement.

A relationship to keep

a + b = (aₓ + bₓ, aᵧ + bᵧ)

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Keep the displacement

    The navy vector a is fixed at (2, 1). The teal vector b starts at the origin; use the sliders to change its components.

  2. STEP 2

    Slide, without rotating

    Press play. Both ends of b move together toward the head of a. Its length, direction, and components stay unchanged throughout this translation.

  3. STEP 3

    Read the resultant

    At the end, the amber arrow connects the origin to the head of the translated b. Its components equal the coordinate sums, even when one component is negative.

Your turn to explain

Make a prediction. Test your reasoning.

If a = (2, 1) and b = (−2, 3), what is a + b?

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

(0, 4). The horizontal changes cancel, while the vertical changes add: 1 + 3 = 4.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Extension · Geometry

Find the part of a displacement along a chosen direction

Use a dot product to split a vector into components parallel and perpendicular to a reference direction.

Undergraduate · Algebra

Describe the same vector in another coordinate system

Solve for coordinates relative to a nonstandard basis.