1.1 Scalars and Vectors in One Dimension

1.1 Scalars and Vectors in One Dimension

A scalar tells you how much of a physical quantity exists, while a vector tells you both how much and in which direction. A car traveling at $20\ \text{m/s}$ and a car traveling at $20\ \text{m/s}$ in the opposite direction have the same speed but different velocities because velocity includes direction.

In one dimension, direction can be represented with a sign. Choose one direction as positive—often rightward or upward—and represent the opposite direction as negative. The sign is not an extra physical quantity; it records the vector’s direction relative to the chosen coordinate axis.

Physical quantities: amount versus direction

A physical quantity is a measurable property described by a number and a unit. Some physical quantities are scalars; others are vectors.

Quantity	Type	Example
Time	Scalar	$t=4.0\ \text{s}$
Mass	Scalar	$m=2.0\ \text{kg}$
Distance	Scalar	$d=15\ \text{m}$
Speed	Scalar	$v=6.0\ \text{m/s}$
Position in one dimension	Vector-like directed quantity	$x=-3.0\ \text{m}$
Displacement	Vector	$\Delta x=+8.0\ \text{m}$
Velocity	Vector	$v=-5.0\ \text{m/s}$
Acceleration	Vector	$a=+2.0\ \text{m/s}^2$

The word vector becomes especially manageable in one dimension because there are only two possible directions along an axis. In two or three dimensions, a vector requires components or an angle; here, a signed number is enough to distinguish opposite directions.

Representing a physical situation

A physical situation becomes easier to reason about when its verbal description is translated into a representation: a diagram, table, chart, or schematic. The representation should make the chosen direction, relevant locations, and measured quantities visible rather than leaving them hidden in prose.

Example: two runners on a straight path. Let east be positive. Runner A is $12\ \text{m}$ east of the starting line, and Runner B is $7\ \text{m}$ west of it.

$$ x_A=+12\ \text{m} $$

$$ x_B=-7\ \text{m} $$

A number-line representation is therefore more informative than two unsigned distances:

$$ \text{west}\qquad\leftarrow\qquad -7\ \text{m}\quad\vert\quad 0\quad\vert\quad +12\ \text{m}\qquad\rightarrow\qquad\text{east} $$

The diagram shows immediately that Runner A is farther east, while Runner B is farther west. It also prevents a common error: treating “$7\ \text{m}$ west” as merely $7\ \text{m}$ and losing the directional information.

Comparing quantities across scenarios

Skill 2.C — Compare physical quantities between two or more scenarios or at different times and locations in a single scenario. Comparison requires identifying both the numerical magnitude and, when applicable, the direction.

Suppose a robot’s position changes from $x_1=-2.0\ \text{m}$ to $x_2=+5.0\ \text{m}$. Its final position has greater magnitude than its initial position:

$$ |x_2|=5.0\ \text{m}>|x_1|=2.0\ \text{m} $$

The robot is also on the opposite side of the origin at the second time. Saying only that “$5.0$ is greater than $2.0$” is incomplete unless the comparison concerns magnitudes. For directed quantities, $+5.0\ \text{m}$ and $-2.0\ \text{m}$ are not simply “larger” and “smaller” versions of the same physical situation; they describe different directions.

Key distinction: A scalar comparison usually concerns magnitude. A vector comparison may require magnitude and direction.

Applying definitions and relationships

Skill 3.B — Apply an appropriate law, definition, theoretical relationship, or model to make a claim. A definition is not just a formula to substitute into; it tells you what physical meaning a calculation represents.

For positions measured along one axis, the displacement definition is

$$ \Delta x=x_f-x_i. $$

For the robot above,

$$ \Delta x=(+5.0\ \text{m})-(-2.0\ \text{m})=+7.0\ \text{m}. $$

The positive result means the robot’s position changed $7.0\ \text{m}$ in the positive direction. This is different from the total distance traveled if the robot changed direction along the way; that distinction becomes important when motion quantities are developed in Topic 1.2.

Science Practice 1: Creating Representations

The course framework uses numbered, color-coded science practices to identify the reasoning emphasized in each lesson. Topics $1.1$ through $1.5$ incorporate Science Practice 1: Creating Representations, Science Practice 2: Mathematical Routines, and Science Practice 3: Scientific Questioning and Argumentation. In this topic, the most direct application is Skill 1.A — Create diagrams, tables, charts, or schematics to represent physical situations.

A strong representation is selective: it includes the coordinate direction, origin, object locations, signs, and units, but does not bury the central relationship under irrelevant decoration. A useful workflow is:

1. Choose and label the positive direction.
2. Mark the origin or reference location.
3. Assign signs to directed quantities.
4. Record numerical values with units.
5. Use the representation to compare or calculate.

Misconception check

Misconception: “A negative vector means the quantity is less than zero physically.” Negative position, velocity, or displacement does not mean that the object has a negative amount of motion or a negative length. It means the vector points opposite the selected positive direction.

Retrieval check: A cyclist is at $x=+10\ \text{m}$ and later at $x=-4\ \text{m}$. Which position has greater magnitude, and what is the cyclist’s displacement?

The position at $x=+10\ \text{m}$ has greater magnitude because $|+10|>|-4|$. The displacement is

$$ \Delta x=(-4\ \text{m})-(+10\ \text{m})=-14\ \text{m}, $$

so the cyclist moved $14\ \text{m}$ in the negative direction.