Concept module

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.

Interactive lab

Loading the live simulation bench.

Progress

Not startedMastery: NewLocal-first

Start exploring and Open Model Lab will keep this concept's progress on this browser first. Challenge mode has 2 compact tasks ready. No finished quick test, solved challenge, or completion mark is saved yet.

Let the live model runChange one real controlOpen What to notice

Try this setup

Jump to a named bench state or copy the one you are looking at now. Shared links reopen the same controls, graph, overlays, and compare context.

Saved setups

Premium keeps named exact-state study setups in your account while stable concept links stay public below.

Checking saved setup access.

This concept can keep using stable links while the saved-setups capability resolves for this browser.

Copy current setup

Stable concept and section links stay public below while exact-state setup sharing stays behind premium.

Stable links

Starter track

Step 1 of 30 / 3 complete

Motion and Circular Motion

Next after this: Projectile Motion.

1. Vectors and Components2. Projectile Motion3. Uniform Circular Motion

This concept is the track start.

Also in Vectors and Motion Bridge.

See next concept

Why it behaves this way

Explanation

A two-dimensional vector does not need a special diagonal rule. You can project it onto the horizontal and vertical axes, then track those perpendicular parts with ordinary algebra and geometry.

This module uses a constant velocity vector so the component idea stays visible. The same magnitude and angle determine the one-second reference vector, the moving point, the component graphs, and the straight-line path, which is why the decomposition carries directly into later mechanics.

Key ideas

01The horizontal and vertical components are projections of the same vector, not separate pushes you invent afterward.

02At fixed magnitude, changing angle redistributes the vector between horizontal and vertical parts.

03If the components stay constant, the resulting motion stays on one straight path while the x(t) and y(t) graphs remain linear.

Frozen walkthrough

Step through the frozen example

Frozen walkthrough

Solve the current vector directly from the live controls. The same magnitude and angle drive the stage, the graphs, and these substitutions, and the time-based example follows the current inspected time unless you freeze it.

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

View plans

Frozen valuesUsing frozen parameters

For the current vector, what horizontal and vertical components does $v$ have?

v

Vector magnitude

8 m/s

θ

Angle

35 °

1. Identify the component relations

Use

v_{x} = v cos (θ)

and

v_{y} = v sin (θ)

2. Substitute the live magnitude and angle

v_{x} = 8 cos (3 5^{\circ})

and

v_{y} = 8 sin (3 5^{\circ})

3. Compute each component

That gives

v_{x} = 6.55 m/s

and

v_{y} = 4.59 m/s

Current components

v_{x} = 6.55 m/s, v_{y} = 4.59 m/s

Both components are positive, so the vector points up and to the right in the first quadrant.

Decompose it

Keep the vector magnitude fixed and rotate the direction closer to the vertical axis. What must happen to the horizontal and vertical components, and how does the straight-line path change?

Make a prediction before you reveal the next step.

Predict which component grows, which shrinks, and whether the path becomes flatter or steeper before you test it.

Check your reasoning against the live bench.

The vertical component grows in magnitude while the horizontal component shrinks, so the path tilts closer to vertical.

At fixed magnitude, rotating the vector redistributes the same total length between the component legs. The resultant stays the same size, but the right-triangle geometry changes.

Common misconception

A diagonal vector needs its own separate rule, so the components are only rough approximations.

The components are exact perpendicular projections of the same vector, which is why the Pythagorean and trigonometric relations recover the original magnitude and angle.

A negative component does not mean the vector got smaller. It only means that piece points leftward or downward relative to the chosen axis.

Quick test

Compare cases

Question 1 of 4

Use the stage, component graph, and position graph together. These checks are about explaining what the vector does, not just naming formulas.

Two vectors have the same magnitude. One points at $3 0^{\circ}$ and the other at $6 0^{\circ}$ . Which comparison is correct?

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

Accessibility

The simulation shows a coordinate plane with a draggable vector anchored at the origin and a point moving in the vector direction over time. Optional overlays show the angle marker, a one-second reference step, and the horizontal and vertical component guides.

Changing the magnitude or angle immediately updates the path, the position graphs, and the constant component graph so the same vector decomposition stays synchronized across every representation.

Graph summary

The path graph is a straight line through the plane because the components stay constant. Hovering or scrubbing the graph moves the stage point to the same place on that line.

The position graph shows linear x(t) and y(t) trends, while the component graph shows flat vx and vy lines because the vector components do not change with time in this model.

Build toward mechanics

Keep this idea moving

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

Turning-effect bridgeMechanics

Vectors and Components

Motion and Circular Motion

Explanation

Step through the frozen example

For the current vector, what horizontal and vertical components does $v$ have?

Two vectors have the same magnitude. One points at $3 0^{\circ}$ and the other at $6 0^{\circ}$ . Which comparison is correct?

Keep this idea moving

Torque

Momentum and Impulse

Projectile Motion

Vectors and Components

Motion and Circular Motion

Explanation

Step through the frozen example

For the current vector, what horizontal and vertical components does v have?

Two vectors have the same magnitude. One points at 30∘ and the other at 60∘. Which comparison is correct?

Keep this idea moving

Torque

Momentum and Impulse

Projectile Motion

For the current vector, what horizontal and vertical components does $v$ have?

Two vectors have the same magnitude. One points at $3 0^{\circ}$ and the other at $6 0^{\circ}$ . Which comparison is correct?