Vectors, Scalars, and Coordinate Systems
Learning Objectives
By the end of this section, you will be able to:- Define and distinguish between scalar and vector quantities.
- Assign a coordinate system for a scenario involving one-dimensional motion.
Figure 1. The motion of this Eclipse Concept jet can be described in terms of the distance it has traveled (a scalar quantity) or its displacement in a specific direction (a vector quantity). In order to specify the direction of motion, its displacement must be described based on a coordinate system. In this case, it may be convenient to choose motion toward the left as positive motion (it is the forward direction for the plane), although in many cases, the x-coordinate runs from left to right, with motion to the right as positive and motion to the left as negative. (credit: Armchair Aviator, Flickr)What is the difference between distance and displacement? Whereas displacement is defined by both direction and magnitude, distance is defined only by magnitude. Displacement is an example of a vector quantity. Distance is an example of a scalar quantity. A vector is any quantity with both magnitude and direction. Other examples of vectors include a velocity of 90 km/h east and a force of 500 newtons straight down.
Some physical quantities, like distance, either have no direction or none is specified. A scalar is any quantity that has a magnitude, but no direction. For example, a 20ºC temperature, the 250 kilocalories (250 Calories) of energy in a candy bar, a 90 km/h speed limit, a person’s 1.8 m height, and a distance of 2.0 m are all scalars—quantities with no specified direction. Note, however, that a scalar can be negative, such as a −20ºC temperature. In this case, the minus sign indicates a point on a scale rather than a direction. Scalars are never represented by arrows.
Coordinate Systems for One-Dimensional Motion
Figure 2. It is usually convenient to consider motion upward or to the right as positive ( + ) and motion downward or to the left as negative (−).Check Your Understanding
A person’s speed can stay the same as he or she rounds a corner and changes direction. Given this information, is speed a scalar or a vector quantity? Explain.Solution
Speed is a scalar quantity. It does not change at all with direction changes; therefore, it has magnitude only. If it were a vector quantity, it would change as direction changes (even if its magnitude remained constant).
Conceptual Questions
1. A student writes, "A bird that is diving for prey has a speed of -10 m/s." What is wrong with the student’s statement? What has the student actually described? Explain.
2. What is the speed of the bird?
3. Acceleration is the change in velocity over time. Given this information, is acceleration a vector or a scalar quantity? Explain.
4. A weather forecast states that the temperature is predicted to be -5ºC the following day. Is this temperature a vector or a scalar quantity? Explain.
Glossary
- scalar:
- a quantity that is described by magnitude, but not direction
- vector:
- a quantity that is described by both magnitude and direction
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