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Math VectorsIntroStarter 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

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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.

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Why it behaves this way

Explanation

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.

Frozen walkthrough

Step through the frozen example

Frozen walkthrough

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

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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.

Make a prediction before you reveal the next step.

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

Check your reasoning against the live bench.

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?

Use the live bench to test the result before moving on.

Accessibility

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 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