Concept module

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.

Interactive lab

Keep the stage, immediate control feedback, and linked study tools in one working bench.

Dot product and projection

Drag A and B on the plane. The amber guide is the projection of B onto A.

Controls

Adjust the live parameters and watch the bench respond.

A x-component

A_{x}

Set the horizontal component of the reference vector A.

A y-component

A_{y}

1.5

Set the vertical component of A.

B x-component

B_{x}

Set the horizontal component of the probe vector B.

B y-component

B_{y}

2.5

Set the vertical component of B.

More tools

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

Show

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

An acute angle gives a positive projection because part of B still points along A.

Graphs

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

Dot product vs angle

Shows how the current magnitudes of A and B turn angle into a positive, zero, or negative dot product.

angle between A and B (°): 0 to 180A dot B: -32 to 32

A dot B

Hover or scrub to link the graph back to the stage.angle between A and B (°) / A dot B

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 part of the reference vector A and changes both the angle marker and the projection direction.

Graph: Dot product vs angleGraph: Projection vs angleOverlay: Angle markerOverlay: Projection guide

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 these prompts to keep the picture, the sign, and the projection interpretation synchronized.

ObservationPrompt 1 of 3

Graph: Dot product vs angle

An acute angle gives a positive projection because part of B still points along A.

Control: B x-componentControl: B y-componentGraph: Dot product vs angleGraph: Projection vs angleOverlay: Angle markerOverlay: Projection guideEquation

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.

2 visible

Overlay focus

Angle marker

Show the angle between A and B at the origin.

What to notice

The sign story changes when the angle passes ninety degrees, not when either vector stops being real.

Why it matters

It makes positive, zero, and negative dot products look like alignment stories instead of separate cases.

Control: A x-componentControl: A y-componentControl: B x-componentControl: B y-componentGraph: Dot product vs angleGraph: Projection vs angleEquation

A_{x}

Equation

A_{y}

Equation

B_{x}

Equation

B_{y}

Challenge mode

Use the angle marker, the amber projection, and the response graph together. This checkpoint is about making orthogonality visible on the same bench instead of naming it after the fact.

0/1 solved

ConditionCore

3 of 5 checks

Orthogonal projection checkpoint

Adjust

B

until the amber projection nearly collapses while both arrows stay clearly nonzero. Keep the angle marker and projection guide visible so the right-angle story stays geometric.

Graph-linkedGuided start2 hints

Suggested start

Keep

A

fixed and rotate

B

until the along-A part disappears rather than shrinking either vector away.

Matched

Open the Dot product vs angle graph.

Dot product vs angle

Matched

Keep the Angle marker visible.

Matched

Keep the Projection guide visible.

Pending

Move the B x-component close to

- 1

Pending

Move the B y-component close to

2

2.5

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

A dot B = 15.75. The angle between the vectors is about 19.25 deg, and the scalar projection of B onto A is 3.69. B still points partly along A, so the projection lands in A's direction.

Equation detailsDeeper interpretation, notes, and worked variable context.

Angle form

Connects the dot product to alignment: acute gives positive, right angle gives zero, and obtuse gives negative.

A \cdot B = ∣ A ∣ ∣ B ∣ cos (θ)

A_{x}

A x-component 4

A_{y}

A y-component 1.5

B_{x}

B x-component 3

B_{y}

B y-component 2.5

Scalar projection

Reads the signed along-A part of B.

comp_{A} (B) = \frac{A \cdot B}{∣ A ∣}

A_{x}

A x-component 4

A_{y}

A y-component 1.5

B_{x}

B x-component 3

B_{y}

B y-component 2.5

Vector projection

Turns the signed along-A amount into an actual vector that lies on A's line.

proj_{A} (B) = \frac{A \cdot B}{∣ A ∣ ^{2}} A

A_{x}

A x-component 4

A_{y}

A y-component 1.5

B_{x}

B x-component 3

B_{y}

B y-component 2.5

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

Short explanation

What the system is doing

The dot product becomes much easier to trust when it stays geometric. This bench keeps two vectors, the angle between them, the signed projection of B onto A, and the scalar A dot B visible together so alignment reads like a picture before it becomes a formula.

The key move is to separate what points along A from what stays perpendicular to A. When B leans with A, the projection is positive. When B turns to ninety degrees, that along-A part collapses. When B leans past ninety degrees, the projection flips and the dot product turns negative for a geometric reason instead of a memorized case.

Key ideas

01The dot product measures alignment by multiplying the size of A by the signed along-A part of B.

02Orthogonal vectors have dot product zero because B contributes no along-A component at all.

03The vector projection of B onto A lies on A's line, while the leftover dashed piece is the perpendicular part.

Worked example

Read the full frozen walkthrough.

Frozen walkthrough

Use these as reading guides while you drag the vectors. The bench still drives the geometry, but the examples stay qualitative so the interpretation remains the focus.

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

How should you read the current dot product from the picture on the bench?

∣ A ∣

Magnitude of A

read from the A card

co m p_{A} (B)

Signed projection of B onto A

read from the projection row

1. Read the size of the reference vector A

Start with the size of A. That sets the scale for how much an along-A component will matter.

2. Read the signed along-A part of B

Then read the signed projection of B onto A. Positive means B still points partly with A, zero means orthogonal, and negative means B points partly against A.

3. Combine size and alignment

The dot product multiplies A's size by that signed along-A part, so the sign comes from alignment while the magnitude scales with how much vector A is available to weight that projection.

Reading rule

A \cdot B = ∣ A ∣ comp_{A} (B)

Use the sign of the projection to decide whether the dot product tells a with-A, orthogonal, or against-A story.

Common misconception

A negative dot product means one of the vectors has a negative length.

Vector lengths stay nonnegative. The sign comes from direction: the projection of B onto A points against A once the angle becomes obtuse.

That is why the same two nonzero vectors can give a positive, zero, or negative dot product without changing how length itself is measured.

Mini challenge

Keep both vectors substantial, then rotate B until the dot product almost vanishes. What must happen to the amber projection as the angle approaches ninety degrees?

Prediction prompt

Predict whether the projection shrinks to zero, keeps the same sign, or flips sign before you test it.

Check your reasoning

The projection shrinks toward zero because B is losing its along-A component.

That is the clean geometric meaning of orthogonality here. B can still be long, but once it points perpendicular to A, none of that length lives along A's direction.

Quick test

Reasoning

Question 1 of 3

Use the angle marker, the amber projection, and the readout together. These checks are about interpreting alignment honestly.

What is the most honest interpretation of a positive dot product on this bench?

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 two draggable vectors from the origin on one coordinate plane. An angle marker can show the current angle between the vectors, and an amber guide can show the projection of B onto A together with the dashed perpendicular leftover.

Changing either vector updates the stage, the readout, and the angle-response graphs together so the learner can compare geometry, sign, and projection without leaving the same bench.

Graph summary

One graph shows how the dot product changes as the angle between the current magnitudes opens from zero to one hundred eighty degrees. The second graph shows how the scalar projection of B onto A changes over that same angle sweep.

Hovering either graph previews the same angle on the stage, so the response curve, the angle marker, and the amber projection remain synchronized.

Carry projection 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 projection language into motion componentsMechanics

Dot Product / Angle and Projection

See each variable before you move it.

Angle marker

Orthogonal projection checkpoint

What the system is doing

Read the full frozen walkthrough.

How should you read the current dot product from the picture on the bench?

What is the most honest interpretation of a positive dot product on this bench?

Keep moving through the concept map.

Vectors and Components

Matrix Transformations / Stretch, Shear, Reflection

Vectors in 2D