Math · Vectors

Dot Product / Angle and Projection

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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: Vectors / ComponentsCarry projection language into motion components.

Key takeaway

A positive dot product means B has a signed along-A part that points with A.
Orthogonal nonzero vectors have dot product zero because B contributes no along-A component.
A negative dot product means the projection of B points against A, not that a vector length became negative.
Scalar and vector projection turn the same alignment idea into a signed length and an arrow on A's line.

Common misconception

Do not read the dot product sign from vector length alone. Read it from the direction of B's projection onto A.

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

Carry projection into motion

Keep this idea moving

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Carry projection language into motion componentsMechanics

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.

Next in Linear actions on the planeVectors

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.

Next in Combination and scalingVectors

Vectors in 2D

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

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Reference and support

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Open reference and support

Equation snapshotAngle form · Scalar projectionShow 3 equations

Read the formulas through the picture: angle controls the sign, scalar projection gives the signed along-A length, and vector projection turns that length into the amber arrow.

Angle form
$A \cdot B = ∣ A ∣ ∣ B ∣ cos (θ)$
Packages alignment into one number: acute gives positive, right angle gives zero, and obtuse gives negative.
Scalar projection
$comp_{A} (B) = \frac{A \cdot B}{∣ A ∣}$
Gives the signed length of the along-A part of B.
Vector projection
$proj_{A} (B) = \frac{A \cdot B}{∣ A ∣ ^{2}} A$
Turns that signed length into a vector that lies on A's line.

Why it behaves this way

Explanation

The dot product is easiest to understand as a measure of alignment. This bench keeps two vectors, the angle between them, the signed projection of $B$ onto $A$ , and the scalar $A \cdot B$ on the same plane so the picture comes first and the formula confirms it.

The key question is: how much of $B$ points along the direction of $A$ ? If that along- $A$ part points with $A$ , the dot product is positive. At $9 0^{\circ}$ , that part is zero. Past $9 0^{\circ}$ , it points against $A$ , so the dot product becomes negative for a geometric reason.

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

has no along-

A

component at all.

03A negative dot product means the projection of

B

points against

A

, not that either vector has a negative length.

Worked examples

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Frozen walkthrough

Step through the frozen example

Frozen walkthrough

Use these as interpretation guides while you drag the vectors. Start by reading the angle and the amber projection, then use the formula to name what the picture is showing.

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Example 1 of 2

Frozen valuesUsing frozen parameters

How can you read the current dot product from the angle and the amber projection on the bench?

∣ A ∣

Magnitude of A

4.27

co m p_{A} (B)

Signed projection of B onto A

3.69

1. Use A as the reference direction

Start with |A| = 4.27. That sets the scale for how much an along-A component will matter.

2. Read how much of B lies along A

Then read comp_A(B) = 3.69. Positive means B still points partly with A, zero means orthogonal, and negative means B points partly against A.

3. Combine size with signed alignment

So A · B = |A| comp_A(B) = 4.27 × 3.69 = 15.75. The sign comes from alignment while the magnitude scales with how much vector A weights that projection.

Dot-product reading

A \cdot B = 15.75

B still keeps an along-A part in A's direction.

Quick test

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Accessibility

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

The simulation shows two draggable vectors from the origin on one coordinate plane. An angle marker can show the angle between them, 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 on one bench.

Graph summary

One graph shows how the dot product changes as the angle between the current vector lengths opens from $0^{\circ}$ to $18 0^{\circ}$ . 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 stay synchronized.

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