2.2 Vectors, Scalars, and Coordinate Systems

Vectors and Scalars

Every measurement in physics falls into one of two categories: scalars or vectors. Knowing which type you're working with determines how you do calculations, so this distinction matters from day one.

• Scalar quantities have magnitude only. Think of mass, temperature, time, and speed. A temperature of 25°C is fully described by that number and its unit.
• Vector quantities have both magnitude and direction. Displacement, velocity, acceleration, and force are all vectors. Saying "5 m/s" isn't enough for velocity; you need to say "5 m/s east."

A common pair that trips students up: speed vs. velocity. Speed is a scalar (just how fast), while velocity is a vector (how fast and in what direction). Speed is actually the magnitude of the velocity vector. The same relationship holds for distance (scalar) and displacement (vector).

A few more examples to lock this in:

• Energy (scalar): a measure of an object's capacity to do work. It has a value in joules but no direction.
• Force (vector): a push or pull described by both its strength and the direction it acts.
• Momentum (vector): the product of mass and velocity, so it inherits velocity's direction.

Coordinate Systems

Coordinate systems give you a framework for measuring where things are and how they move. Without one, terms like "positive velocity" or "negative displacement" have no meaning.

One-Dimensional Motion

For motion along a straight line, a single axis is all you need. Here's how to set one up:

1. Choose an origin. This is your reference point where position equals zero.
2. Define the positive direction. By convention, rightward or upward is usually positive, but you can choose whatever makes the problem easier.
3. The negative direction is automatically the opposite of your positive choice.

Position ($x$) describes where an object is relative to the origin. Displacement ($\Delta x$) describes the change in position:

$\Delta x = x_f - x_i$

If $\Delta x$ is positive, the object moved in the positive direction. If $\Delta x$ is negative, it moved in the negative direction. The same sign convention applies to velocity and acceleration along that axis.

Two-Dimensional Coordinate Systems

When motion isn't confined to a line, you need more than one axis.

• Cartesian coordinates use perpendicular axes ($x$ and $y$, or $x$, $y$, and $z$ in three dimensions). Each axis is independent, which makes this system great for breaking problems into parts.
• Polar coordinates describe a point using a distance from the origin ($r$) and an angle ($\theta$). These show up later in circular motion problems.

For this course, you'll mostly work in Cartesian coordinates.

Vector Notation

Vectors are written with an arrow above the symbol: $\vec{v}$ for velocity, $\vec{a}$ for acceleration, $\vec{F}$ for force. In textbooks, they're sometimes printed in bold instead.

The magnitude of a vector is its size without any direction, written as $|\vec{v}|$ or simply $v$. For example, if $\vec{v} = 5 \text{ m/s, east}$, then $|\vec{v}| = 5 \text{ m/s}$.

Direction can be stated in words ("east," "30° above horizontal") or through components along coordinate axes:

$\vec{F} = F_x \hat{i} + F_y \hat{j}$

Here, $\hat{i}$ and $\hat{j}$ are unit vectors pointing in the positive $x$ and $y$ directions, respectively. $F_x$ and $F_y$ are the projections of $\vec{F}$ onto those axes.

To find the magnitude of a vector from its components, use the Pythagorean theorem:

$|\vec{F}| = \sqrt{F_x^2 + F_y^2}$

For example, if $F_x = 3 \text{ N}$ and $F_y = 4 \text{ N}$, then $|\vec{F}| = \sqrt{9 + 16} = 5 \text{ N}$.

Vector addition combines multiple vectors into a single resultant vector. In component form, you just add the corresponding components: the $x$-components together and the $y$-components together. You'll use this constantly once you start working with forces and projectile motion.