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MathVectorsIntroStarter track

Concept module

Vectors in 2D

Combine, subtract, and scale vectors on one plane so magnitude, direction, and components stay tied to the same live object.

Interactive lab

Keep the stage, graph, and immediate control feedback in one working view.

Vectors in 2D

sA + B

Drag A and B on the plane. Use the scalar to stretch or flip A.

Graphs

Switch graph views without breaking the live stage and time link.

Result components vs scalar

Shows how the x- and y-components of the resultant change as the scalar on A varies.

scalar s: -2.5 to 2.5component value: -16 to 16

result xresult y

Hover or scrub to link the graph back to the stage.scalar s / component value

Controls

Adjust the physical parameters and watch the motion respond.

A x-component

A_{x}

Set the horizontal component of vector A.

A y-component

A_{y}

Set the vertical component of vector A.

B x-component

B_{x}

1.5

Set the horizontal component of vector B.

B y-component

B_{y}

Set the vertical component of vector B.

More tools

Secondary controls, alternate presets, and less-used toggles stay nearby without crowding the main bench.

Hide

Scalar on A

s

Stretch, compress, or flip A before it combines with B.

Subtract B

-

Show subtraction as adding the opposite vector.

More presets

Presets

Predict -> manipulate -> observe

Keep the active prompt next to the controls so each change has an immediate visible consequence.

ObservationPrompt 1 of 3

Drag either endpoint and compare the arrow on the plane with the ordered-pair readout. They are two views of the same vector.

Equation map

See each variable before you move it.

Select a symbol to highlight the matching control and the graph or overlay it most directly changes.

A_{x}

A x-component

Moves the horizontal component of vector A while keeping the same geometric arrow on the plane.

Graph: Result components vs scalarGraph: Result magnitude vs scalarOverlay: Scaled AOverlay: Tip-to-tail view

Equations in play

Choose an equation to sync the active symbol, control highlight, and related graph mapping.

More tools

Detailed noticing prompts, guided overlays, and challenge tasks stay available without taking over the main bench.

Hide

What to notice

Use the live prompts to keep the plane picture and the component algebra locked together.

ObservationPrompt 1 of 3

Graph: Result components vs scalar

Drag either endpoint and compare the arrow on the plane with the ordered-pair readout. They are two views of the same vector.

Control: A x-componentControl: A y-componentControl: B x-componentControl: B y-componentGraph: Result components vs scalarOverlay: Component guidesEquation

A_{x}

Equation

A_{y}

Equation

B_{x}

Equation

B_{y}

Guided overlays

Focus one overlay at a time to see what it represents and what to notice in the live motion.

3 visible

Overlay focus

Component guides

Project the resultant onto the x- and y-axes.

What to notice

The same arrow can be read geometrically on the plane and algebraically through its components.

Why it matters

It keeps the coordinate and geometric views of the resultant synchronized.

Control: A x-componentControl: A y-componentControl: B x-componentControl: B y-componentControl: Scalar on AGraph: Result components vs scalarEquation

A_{x}

Equation

A_{y}

Equation

B_{x}

Equation

B_{y}

Challenge mode

Use the plane like a real vector-combination bench. The goal is to make component cancellation visible while the vectors themselves stay substantial.

0/1 solved

MatchCore

6 of 7 checks

Near-zero resultant

Adjust the vectors until the resultant lands very close to the origin, but keep the scaled first vector clearly nontrivial so the cancellation has to be earned.

Graph-linkedGuided start2 hints

Suggested start

Use the tip-to-tail view and the component guides together while you cancel the x- and y-components.

Matched

Open the Result components vs scalar graph.

Result components vs scalar

Matched

Keep the Component guides visible.

Matched

Keep the Tip-to-tail view visible.

Matched

Keep the Scaled A visible.

Pending

Bring the resultant magnitude below

0.35

6.73

Matched

Keep the scaled first vector magnitude between

3.3

and

3.9

3.61

Matched

Keep the scalar close to

1

so the cancellation comes mostly from vector opposition, not from shrinking A away.

The checklist updates from the live simulation state, active graph, overlays, inspect time, and compare setup.

sA + B gives result <4.5, 5>. The scalar keeps vector A at its original size before combination. The result points at about 48.01 degrees with magnitude 6.73.

Equation detailsDeeper interpretation, notes, and worked variable context.

Component form

Records each vector as an ordered pair on the same 2D plane.

A = ⟨ A_{x}, A_{y} ⟩, B = ⟨ B_{x}, B_{y} ⟩

A_{x}

A x-component 3

A_{y}

A y-component 2

B_{x}

B x-component 1.5

B_{y}

B y-component 3

Resultant rule

Shows that addition and subtraction are both component-wise once the effective second vector is chosen.

r = s A \pm B = ⟨ s A_{x} \pm B_{x}, s A_{y} \pm B_{y} ⟩

A_{x}

A x-component 3

A_{y}

A y-component 2

B_{x}

B x-component 1.5

B_{y}

B y-component 3

s

Scalar 1

-

Subtract mode Off

Resultant magnitude

Reconstructs the vector length from the perpendicular components.

∣ r ∣ = r_{x}^{2} + r_{y}^{2}

s

Scalar 1

Progress

Not startedMastery: NewLocal-first

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Let the live model runChange one real controlOpen What to notice

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Stable links

Starter track

Step 1 of 20 / 2 complete

Vectors and Motion Bridge

Next after this: Vectors and Components.

1. Vectors in 2D2. Vectors and Components

This concept is the track start.

See next concept

Short explanation

What the system is doing

A 2D vector becomes easier to trust when the algebra and geometry stay on the same plane. This module keeps two vectors, a scalar multiplier, and the resultant visible together so addition, subtraction, and scaling all read as actual movement on one coordinate system.

The important habit is to treat a vector as both a whole arrow and an ordered pair of components. The plane shows the direction and length, while the component readout and response graphs show the same object in algebraic form.

Key ideas

01A vector in 2D has both magnitude and direction, and its components record the same object in coordinate form.

02Vector addition and subtraction can be read geometrically with tip-to-tail arrows or algebraically by combining components.

03Scalar multiplication changes the size of a vector, and a negative scalar flips its direction through the origin.

Worked example

Read the full frozen walkthrough.

Frozen walkthrough

Use the current vectors and scalar from the plane. The same live state drives the stage, the response graphs, and these substitutions.

Live worked examples are available on Premium. You can still read the full frozen walkthrough on the free tier.

View plans

Frozen valuesUsing frozen parameters

For the current setup, what components does the resultant vector have?

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>.

Current 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.

Common misconception

Vector subtraction needs a separate geometric rule from vector addition.

Subtraction is still addition, but with the opposite vector.

That is why the same tip-to-tail picture works once B is reversed to -B.

Mini challenge

Adjust the vectors until the resultant is small in magnitude even though neither vector is small by itself.

Prediction prompt

Decide whether you need stronger cancellation from the second vector, a flipped scalar, or both before you test it.

Check your reasoning

You need the combined x- and y-components to nearly cancel so the resultant lands close to the origin.

That is the cleanest way to see that vector addition is component bookkeeping and geometry at the same time. The arrows can look substantial while the net result stays small if their components oppose each other.

Quick test

Reasoning

Question 1 of 3

Use the plane, the component readout, and the response graphs together. These checks are about making the geometric and algebraic views agree.

Which statement is the cleanest description of a 2D vector?

Choose one answer to reveal feedback, then test the idea in the live system if a guided example is available.

Accessible description

The simulation shows a coordinate plane with two draggable vectors from the origin, a scaled version of the first vector, and a resultant vector. Optional overlays show the tip-to-tail construction, the resultant component guides, and the scaled first vector.

Changing any component, the scalar, or subtract mode updates the plane, the algebraic readout, and the response graphs together so the learner can compare the geometric and algebraic views directly.

Graph summary

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

Those graphs are tied to the same live plane, so the current scalar value and the current resultant arrow match the highlighted response point.

Bridge vectors into motion

Keep moving through the concept map.

These suggestions come from the concept registry, so the reason label reflects either curated guidance or the fallback progression logic.

Carry the same vector language into motionMechanics

Vectors in 2D

See each variable before you move it.

Component guides

Near-zero resultant

Vectors and Motion Bridge

What the system is doing

Read the full frozen walkthrough.

For the current setup, what components does the resultant vector have?

Which statement is the cleanest description of a 2D vector?

Keep moving through the concept map.

Vectors and Components

Projectile Motion

Integral as Accumulation / Area