Math · Vectors

Vectors in 2D

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Wrap-up

What you learned

Recommended next: Open concept testCheck whether the core ideas are ready without leaving this concept.
Read next: Matrix TransformationsLet matrices move the same plane, basis vectors, and arrows.

Key takeaway

A 2D vector can be read as one arrow and as one ordered pair of components.
Vector addition and subtraction share the same tip-to-tail picture once subtraction is treated as adding the opposite vector.
A scalar changes A before the combination step; a negative scalar flips its direction.
A small resultant is convincing only when the x- and y-components both explain the cancellation.

Common misconception

Do not treat the plane picture and component arithmetic as separate methods. They are two descriptions of the same vector state.

Subtraction is still vector addition, but with the opposite of the second vector.

Bridge vectors into motion

Keep this idea moving

Open the next concept, route, or track only when you want the current model to widen into a larger branch.

Turn the same plane into a matrix action on the grid and basis vectorsVectors

Matrix Transformations / Stretch, Shear, Reflection

Let one 2 by 2 matrix act on a grid, the basis vectors, and a sample shape so stretch, shear, reflection, and combined plane changes stay visual instead of symbolic-only.

Turn combination into alignment and projectionVectors

Dot Product / Angle and Projection

Keep two vectors, their angle, the signed projection of one onto the other, and the dot product visible together so alignment reads geometrically instead of as memorized cases.

Carry the same vector language into motionMechanics

Vectors and Components

Rotate and scale a live vector, decompose it into horizontal and vertical parts, and watch those components drive the same straight-line motion and geometry.

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More practice optionsReview on the bench · Worked examples

Practice optionReview on the benchReplay the guided setup with the key controls and graphs in sight.
Practice optionWorked examplesCompare against solved walkthroughs after you try the setup yourself.
Practice optionFree play on the benchExperiment freely with the live controls before moving to another concept.

Reference and support

Use these calmer sections when you want a fuller explanation, examples, or accessibility support after the guided flow.

Return here when you want the slower reference pass.

Open reference and support

Equation snapshotComponent form · Resultant ruleShow 3 equations

Read the component rule as the algebra version of the arrows: scale A first, choose B or -B, then combine x with x and y with y.

Component form
$A = ⟨ A_{x}, A_{y} ⟩, B = ⟨ B_{x}, B_{y} ⟩$
Records each vector as an ordered pair on the same 2D plane.
Resultant rule
$r = s A \pm B = ⟨ s A_{x} \pm B_{x}, s A_{y} \pm B_{y} ⟩$
Shows that addition and subtraction are both component-wise once the effective second vector is chosen.
Resultant magnitude
$∣ r ∣ = r_{x}^{2} + r_{y}^{2}$
Reconstructs the vector length from the perpendicular components.

Why it behaves this way

Explanation

A 2D vector is easiest to understand when you can see both the arrow and its components at the same time. This module keeps A, B, the scaled vector sA, and the resultant on one plane so addition, subtraction, and scaling can all be read from one shared picture.

The key habit is to switch between arrow language and component language without treating them as different objects. When a vector changes on the plane, its ordered pair and the graphs change with it, so the geometry and the algebra should always agree.

Key ideas

01A 2D vector is one object with two equivalent descriptions: an arrow on the plane and an ordered pair of components.

02Addition and subtraction both work component by component, and the same tip-to-tail picture still works if subtraction is treated as adding the opposite vector.

03A scalar changes the size of A, and a negative scalar also reverses its direction through the origin.

Worked examples

Open examples when you want to see the same idea walked through step by step.

Frozen walkthrough

Step through the frozen example

Frozen walkthrough

Use the current vectors and scalar from the plane. The same live state drives the arrows, the graphs, and these substitutions, so every step should match something you can see.

Supporter unlocks saved study tools, exact-state sharing, and the richer review surfaces that support this guided flow.

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Example 1 of 2

Frozen valuesUsing frozen parameters

For the current setup, what are the x- and y-components of the resultant vector?

s

Scalar

A_{x}

A x-component

A_{y}

A y-component

B_{x}

B x-component

1.5

B_{y}

B y-component

1. Scale vector A first

The scaled first vector is sA = <3, 2>.

2. Combine it with the effective second vector

The current operation is sA + B, so the second piece contributes <1.5, 3>.

3. Add the components

That gives result = <3 + 1.5, 2 + 3> = <4.5, 5>.

Live resultant

r = ⟨ 4.5, 5 ⟩

Addition is being shown tip-to-tail, so the resultant runs from the origin to the final endpoint after sA and B are combined.

Quick test

Loading saved test state.

Accessibility

Open the text-first descriptions when you need the simulation and graph translated into words.

The simulation shows a coordinate plane with two vectors from the origin, a scaled copy of A, and a resultant vector. Optional overlays can show the tip-to-tail construction, the resultant's component guides, and the scaled copy of A before combination.

Changing any component, the scalar, or subtract mode updates the plane, the algebraic readout, and the graphs together, so the geometric and component views stay synchronized.

Graph summary

One graph shows how the resultant's x- and y-components change as the scalar on A changes. The other shows how the resultant magnitude changes over the same scalar scan.

The highlighted point on each graph matches the current plane setup, so you can connect the current scalar setting to one exact resultant arrow.

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Progress

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Starter track

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Vectors and Motion Bridge

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