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.

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 your initial to your 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.
• Displacement can be zero even after motion. If you walk 50 m in a loop back to your starting point, $\vec{d} = 0$ even though you covered 50 m of distance (a scalar).

Velocity

• Rate of change of displacement, defined as $\vec{v} = \frac{\Delta \vec{d}}{\Delta t}$, combining speed with direction
• Average velocity uses total displacement over total time, while instantaneous velocity is the limit as the time interval shrinks: $\vec{v} = \frac{d\vec{x}}{dt}$
• Velocity changes whenever direction changes. An object moving in a circle at constant speed still has changing velocity because the direction of motion is continuously shifting.

Acceleration

• Rate of change of velocity, defined as $\vec{a} = \frac{\Delta \vec{v}}{\Delta t}$, measuring how quickly velocity changes in magnitude, direction, or both
• Acceleration can point opposite to motion. When $\vec{a}$ opposes $\vec{v}$, the object slows down. This is often called deceleration, but it's still just acceleration in the opposite direction.
• Centripetal acceleration points toward the center of a circular path, changing the direction of velocity without changing its magnitude (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 mass in motion.

Force

• An interaction that changes an object's 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: $\vec{F}_{net} = m\vec{a}$. The net force is the vector sum of all individual forces acting on the object.
• Forces applied at angles must be resolved into components (typically $x$ and $y$) before you can add them. You find the net force by summing components separately: $F_{net,x} = \sum F_x$ and $F_{net,y} = \sum F_y$.

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 an event equals total momentum after, as long as no external net force acts on the system
• Impulse changes momentum. The impulse-momentum theorem states $\vec{J} = \Delta \vec{p} = \vec{F}_{avg} \Delta t$. A large force over a short time and a small force over a long time can deliver the same impulse.

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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 the fingers of your right hand in the direction of rotation, and your thumb points along $\vec{\omega}$. This means angular velocity points along the axis of rotation, not along the circular path.
• Links to linear (tangential) velocity through $v = r\omega$, where $r$ is the distance from the rotation axis. Points farther from the axis move faster even though they share the same $\omega$.

Torque

• The rotational equivalent of force, measuring the tendency to cause angular acceleration. Defined as $\vec{\tau} = \vec{r} \times \vec{F}$, where $\vec{r}$ is the position vector from the pivot to the point where the force is applied.
• Magnitude depends on the lever arm and angle: $\tau = rF\sin(\theta)$, where $\theta$ is the angle between $\vec{r}$ and $\vec{F}$. Maximum torque occurs when the force is perpendicular to the lever arm ($\theta = 90°$, so $\sin(\theta) = 1$). Zero torque occurs when the force is parallel to $\vec{r}$ ($\theta = 0°$).
• Rotational equilibrium requires zero net torque: $\sum \vec{\tau} = 0$. Note that an object can have $\sum \vec{F} = 0$ (no translational acceleration) but still have a net torque causing it to spin.

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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 (in the magnetic case) moving charge.

Electric Field

• Force per unit positive charge, defined as $\vec{E} = \frac{\vec{F}}{q}$, measured in newtons per coulomb (N/C) or equivalently volts per meter (V/m)
• Field lines point away from positive charges and toward negative charges. The direction shows which way a positive test charge would accelerate if placed in the field.
• Superposition applies: the net field from multiple charges is the vector sum of the individual fields from each charge.

Magnetic Field

• Influences moving charges and currents, measured in teslas (T), where $1 \text{ T} = 1 \frac{\text{N}}{\text{A} \cdot \text{m}}$
• The force on a moving charge is perpendicular to both the velocity and the field. The Lorentz force law gives $\vec{F} = q\vec{v} \times \vec{B}$, and you use the right-hand rule to find the direction. Because the force is always perpendicular to velocity, magnetic forces do no work and can't change a particle's speed, only its direction.
• There are no magnetic monopoles. Magnetic field lines always form closed loops, running from north to south outside a magnet and south to north inside it.

Gravitational Field

• Force per unit mass, defined as $\vec{g} = \frac{\vec{F}_g}{m}$, measured in N/kg (which is dimensionally 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. Double your distance from the center of a planet, and the gravitational field strength drops to one-quarter.

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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. Practice problem: 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.