Physics · Mechanics

Vectors and Components

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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: Projectile MotionUse horizontal and vertical components to predict a curved path.

Key takeaway

A vector component is an exact projection of one vector onto an axis.
Changing angle at fixed magnitude redistributes the same vector between horizontal and vertical components.
Component signs encode direction along the axes, while the resultant magnitude comes from the perpendicular legs.
Constant components produce straight-line motion with matching x(t), y(t), path, and stage representations.

Common misconception

Do not treat components as two new vectors added after the fact; they are the axis shadows of the original vector.

The components are exact perpendicular projections of the same vector. Using trigonometry and the Pythagorean relation takes you back to the original magnitude and angle.

Build toward mechanics

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Turning-effect bridgeMechanics

Torque

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Momentum linkMechanics

Momentum and Impulse

Push one cart with a timed force pulse and watch momentum, impulse, and force-time area stay tied to the same motion, readouts, and graphs.

Trajectory applicationMechanics

Projectile Motion

Launch a projectile, watch the trajectory form, and connect the range, height, and component motion to the launch settings.

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Reference and support

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Equation snapshotComponent snapshot · Vertical componentShow 3 equations

Use the angle to split one vector into exact x and y projections, then use the same components to predict the straight-line motion.

Component snapshot
$v_{x} = v cos (θ)$
Read as: horizontal component equals vector magnitude times cosine of the angle
Projects the vector onto the horizontal axis.
Vertical component
$v_{y} = v sin (θ)$
Read as: vertical component equals vector magnitude times sine of the angle
Projects the vector onto the vertical axis.
Resultant magnitude
$v = v_{x}^{2} + v_{y}^{2}$
Reconstructs the original vector from its perpendicular components.

Why it behaves this way

Explanation

A diagonal vector is not a special new object. It is one vector split into a horizontal part and a vertical part, so you can analyze it by tracking two perpendicular components.

In this model the vector stays constant, so the component graph stays flat, the $x (t)$ and $y (t)$ graphs stay linear, and the point follows a straight path. One magnitude and one angle control every representation, which is why component thinking carries directly into later mechanics.

Key ideas

01The horizontal and vertical components are exact projections of one vector, not two separate motions you invent afterward.

02At fixed magnitude, changing angle redistributes the same vector between

v_{x}

and

v_{y}

03If

v_{x}

and

v_{y}

stay constant, the point moves on a straight line, the component graph stays flat, and

x (t)

and

y (t)

remain linear.

Worked examples

Work the live vector

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Frozen walkthrough

Step through the frozen example

Frozen walkthrough

Read the current magnitude and angle from the controls, then check the calculation against the stage and graphs. In the time-based example, the values follow the current inspected time unless you freeze it.

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Example 1 of 2

Frozen valuesUsing frozen parameters

For the current vector, what are $v_{x}$ and $v_{y}$ ?

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

Live 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.

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Motion and Circular Motion

Projectile Motion is what this track opens next.

This concept is the track start.

Also appears in Vectors and Motion Bridge.

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