AP Physics 1 · Unit 1 · Topic 1.1
Scalars and Vectors in One Dimension
Motion along a straight line still requires a direction convention. A scalar answers “how much?” A vector answers both “how much?” and “which way?” In one dimension, an arrow diagram and a signed component are two representations of the same vector model.
Learning Goals
After this lesson, you should be able to:
- classify a physical quantity as scalar or vector and justify the classification;
- choose an origin and positive direction for motion on one axis;
- translate between a vector arrow, a directional statement, and a signed component;
- add collinear vectors and interpret the sign and magnitude of the result; and
- distinguish distance traveled from displacement.
1. Magnitude, Direction, and Sign
A scalar is completely described by a magnitude and unit. A vector also requires direction. The magnitude of a vector is nonnegative; a minus sign on a one-dimensional component reports direction relative to the chosen axis, not a “negative size.”
The arrow over \(\Delta\vec{x}\) marks a vector. Once one axis has been chosen, the component \(\Delta x\) can be written without an arrow because its sign already encodes direction.
2. Classifying Common Quantities
| Quantity | Type | What completes the description? |
|---|---|---|
| Distance | Scalar | Total path length and unit |
| Speed | Scalar | Magnitude and unit |
| Position | Vector in this model | Location relative to an origin and direction |
| Displacement | Vector | Change in position with direction |
| Velocity | Vector | Rate and direction of motion |
| Acceleration | Vector | Rate and direction of velocity change |
A signed scalar can still be a scalar. For example, a temperature below zero has a sign but no spatial direction. Classification depends on physical meaning, not merely on whether a number has a plus or minus sign.
3. Build the Coordinate System First
- Name the object or system being tracked.
- Choose an origin, \(x=0\).
- Choose one direction as positive and label it on the diagram.
- Assign signs to every vector component using that same convention.
- Keep the convention fixed through the entire solution.
Rightward does not have to be positive. Either choice produces the same physical conclusion when it is used consistently; only the component signs change.
4. One Axis, Three Representations
5. Vector Addition in One Dimension
Choose a positive direction, convert each collinear vector into a signed component, and add algebraically. Vectors in the positive direction contribute positive components; vectors in the opposite direction contribute negative components.
The sign of the sum gives its direction. Its magnitude is \(|\Delta x_{\mathrm{net}}|\). Do not add magnitudes alone unless every vector points in the same direction.
6. Worked Example: Trip with a Reversal
Question: A student begins at \(x_i=-3\,\mathrm m\), walks \(12\,\mathrm m\) to the right, and then \(5\,\mathrm m\) to the left. Take right as positive. Find the final position, displacement, and distance traveled.
Represent: \(\Delta x_1=+12\,\mathrm m\) and \(\Delta x_2=-5\,\mathrm m\).
Answer: The student finishes at \(+4\,\mathrm m\), has displacement \(+7\,\mathrm m\) (7 m right), and travels a distance of \(17\,\mathrm m\). Distance and displacement differ because the path reverses direction.
7. AP Reasoning Example: Same Magnitude, Different Result
Cart A undergoes two successive displacements of \(+4\,\mathrm m\) and \(+3\,\mathrm m\). Cart B undergoes \(+4\,\mathrm m\) and \(-3\,\mathrm m\). Both carts travel \(7\,\mathrm m\), but their net displacements are different.
Claim: Equal distance does not imply equal displacement. Evidence: the same two path-length magnitudes produce different signed sums. Reasoning: distance ignores direction, whereas displacement is a vector change in position.
8. Investigation: Human Number Line
Place tape marks at one-meter intervals along a straight floor. Choose one mark as the origin and declare a positive direction. For several walks that include a reversal, record \(x_i\), every signed displacement component, \(x_f\), total distance, and net displacement.
Consistency tests:
Repeat one walk after reversing the positive axis. The component signs should reverse, while the path, distance, and physical endpoint remain unchanged.
9. Common Traps and Fast Checks
- Trap: treating a negative component as a negative magnitude. Check: state the direction in words.
- Trap: adding \(x_i\) and \(x_f\) to find displacement. Check: use final minus initial.
- Trap: calculating distance from \(|x_f-x_i|\) after a reversal. Check: add the length of every path segment.
- Trap: changing the positive direction halfway through. Check: draw and retain one labeled axis.
- Trap: assuming any negative number is a vector. Check: ask whether spatial direction is part of the quantity's definition.
10. Key Takeaways
- Scalars have magnitude; vectors have magnitude and direction.
- In one dimension, a component sign fully identifies one of the two axis directions.
- The vector magnitude is nonnegative even when its component is negative.
- Displacement is final position minus initial position; distance is total path length.
- A coordinate choice is arbitrary, but consistency is mandatory.
A cart moves \(6\,\mathrm m\) left and then \(10\,\mathrm m\) right. If left is positive, write both signed displacement components, find their vector sum, and explain why changing the positive direction would not change the cart's physical endpoint.
Concept scope cross-checked with the College Board AP Physics 1 course framework and OpenStax College Physics 2e, Section 2.2. Explanations, diagram, examples, and investigation are independently written for Insomnia Studyroom.
Official curriculum reference: College Board AP Physics 1 course page. The explanation and worked example are independently written for this study site.