Vector Quantities in Physics

Why This Matters

In physics, the difference between scalar and vector quantities isn't just a vocabulary distinction—it's the key to solving nearly every mechanics problem you'll encounter. Vectors carry both magnitude (how much) and direction (which way), and understanding this dual nature is what separates students who can plug into formulas from those who actually understand why objects move the way they do. You're being tested on your ability to add vectors, resolve them into components, and recognize when direction changes even if speed doesn't.

The vector quantities covered here form an interconnected web: displacement leads to velocity, velocity changes give you acceleration, forces cause that acceleration, and momentum tracks the result. Whether you're analyzing a projectile's curved path, calculating net force, or predicting collision outcomes, you need to think in vectors. Don't just memorize definitions—know what physical change each quantity describes and how direction affects your calculations.

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Motion Vectors: Describing Where and How Fast

These quantities describe an object's position and how that position changes over time. They form the foundation of kinematics—the study of motion without considering its causes.

Displacement

• Change in position from start to finish—the straight-line vector from initial to final location, regardless of the path taken
• Direction matters as much as distance; two objects traveling 10 m end up in different places if they go opposite directions
• Can be zero even after motion—returning to your starting point means $\vec{d} = 0$ even if you traveled a long path

Velocity

• Rate of change of displacement—defined as $\vec{v} = \frac{\Delta \vec{d}}{\Delta t}$, combining speed with direction
• Average vs. instantaneous; average velocity uses total displacement over total time, while instantaneous velocity is the derivative $\frac{d\vec{x}}{dt}$
• Changes when direction changes—an object moving in a circle at constant speed still has changing velocity because the direction changes

Acceleration

• Rate of change of velocity—defined as $\vec{a} = \frac{\Delta \vec{v}}{\Delta t}$, measuring how quickly velocity changes
• Can point opposite to motion; negative acceleration (deceleration) occurs when $\vec{a}$ opposes $\vec{v}$, slowing the object down
• Centripetal acceleration points toward the center of a circular path, changing direction without changing speed

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Cause Vectors: What Makes Motion Change

These quantities explain why motion changes. Force causes acceleration, and momentum tracks the cumulative effect of forces over time.

Force

• Interaction that changes motion—measured in newtons (N), where $1 \text{ N} = 1 \text{ kg} \cdot \text{m/s}^2$
• Newton's second law connects force to acceleration: $\vec{F}_{net} = m\vec{a}$, making net force the vector sum of all forces
• Direction determines outcome; forces at angles must be resolved into components before adding

Momentum

• Product of mass and velocity—defined as $\vec{p} = m\vec{v}$, representing an object's "quantity of motion"
• Conserved in isolated systems; total momentum before equals total momentum after when no external forces act
• Impulse changes momentum; the impulse-momentum theorem states $\vec{J} = \Delta \vec{p} = \vec{F}_{avg} \Delta t$

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Rotational Vectors: Spinning and Turning

These quantities extend linear concepts to rotation. Angular velocity describes spinning motion, while torque describes what causes that rotation to change.

Angular Velocity

• Rate of rotation about an axis—measured in radians per second (rad/s), defined as $\vec{\omega} = \frac{\Delta \theta}{\Delta t}$
• Direction follows the right-hand rule; curl fingers in the direction of rotation, and your thumb points along $\vec{\omega}$
• Links to linear velocity through $v = r\omega$, connecting tangential speed to rotation rate

Torque

• Rotational equivalent of force—measures the tendency to cause angular acceleration, defined as $\vec{\tau} = \vec{r} \times \vec{F}$
• Magnitude depends on lever arm; $\tau = rF\sin(\theta)$, where $\theta$ is the angle between $\vec{r}$ and $\vec{F}$
• Equilibrium requires zero net torque; for rotational equilibrium, $\sum \vec{\tau} = 0$, not just $\sum \vec{F} = 0$

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Field Vectors: Forces at a Distance

Fields describe how forces act through space without direct contact. Each field type represents force per unit of something—charge, mass, or current.

Electric Field

• Force per unit positive charge—defined as $\vec{E} = \frac{\vec{F}}{q}$, measured in newtons per coulomb (N/C) or volts per meter (V/m)
• Points away from positive, toward negative; field lines show the direction a positive test charge would accelerate
• Superposition applies; the net field from multiple charges is the vector sum of individual fields

Magnetic Field

• Influences moving charges and currents—measured in teslas (T), where $1 \text{ T} = 1 \frac{\text{N}}{\text{A} \cdot \text{m}}$
• Force perpendicular to both velocity and field; the Lorentz force is $\vec{F} = q\vec{v} \times \vec{B}$, requiring the right-hand rule
• No magnetic monopoles; field lines form closed loops, always going from north to south poles externally

Gravitational Field

• Force per unit mass—defined as $\vec{g} = \frac{\vec{F}_g}{m}$, measured in N/kg (equivalent to m/s²)
• Always points toward the source mass; near Earth's surface, $\vec{g}$ points downward with magnitude $\approx 9.8 \text{ m/s}^2$
• Decreases with distance squared; $g = \frac{GM}{r^2}$, following the inverse-square law

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Quick Reference Table

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Self-Check Questions

1. Which two motion vectors are related by a time derivative? Explain how you would find one from the other mathematically.

2. Compare and contrast momentum and kinetic energy. Both depend on mass and velocity—why is momentum a vector while kinetic energy is a scalar?

3. An object moves in a circle at constant speed. Which vector quantities are changing, and which remain constant? Explain why.

4. FRQ-style: A force is applied at an angle to a wrench. What determines whether this produces more or less torque than a force applied perpendicular to the wrench? Write the relevant equation and explain each term.

5. All three field vectors (electric, magnetic, gravitational) describe "force per unit something." Identify what that "something" is for each field, and explain why the magnetic force equation requires velocity while the others don't.