Problem 93 - Precalculus

Question

Would a scalar or a vector represent the following? The car is driving 72 mph due east \((90^{\circ}\) with respect to north).

Step-by-Step Solution

Answer

The description 'driving 72 mph due east' represents a vector.

Step 1: Understanding Scalars and Vectors

A scalar is a physical quantity only described by magnitude, such as temperature or speed. A vector is a physical quantity that has both magnitude and direction, such as force or velocity.

Step 2: Analyzing the Given Information

The problem states: "The car is driving 72 mph due east." It specifies both the speed (magnitude) and the direction (east, 90 degrees with respect to north).

Step 3: Determining the Representation Type

Since the information includes both a magnitude (72 mph) and a direction (due east), it is characterized by a vector. Vectors are used to represent physical quantities that have both magnitude and direction, such as velocity.

Key Concepts

Scalars

Scalars are straightforward because they are quantities that describe only how much or how many. They do not provide any sense of direction. For example, numbers like temperature, mass, and speed are scalars. Imagine asking only "how fast?" — the answer, such as 72 miles per hour (mph), gives you the speed but nothing about where that speed is heading.

• Magnitude: Scalars have only magnitude. This means they have size or amount but nothing else.
• No Direction: Scalars lack directional information. In this context, direction is irrelevant and absent.

Understanding scalars is crucial because they often serve as foundational pieces for constructing more complex concepts like vectors. They tell us part of the story, focusing solely on quantity.

Velocity

Velocity is a vector quantity, meaning it boasts both magnitude and direction. Think of velocity as enhanced speed. It not only answers "how fast?" but also "where to?" For instance, if a car travels at 72 mph directly east, this is velocity.

• Magnitude with Direction: Velocity combines the speed of an object with the direction it is heading.
• Examples of Velocity: A train moving 100 km/h north, or a bird flying 20 m/s southwest, specify both speed and direction, hence velocity.

Concisely, while speed tells you how fast something moves, velocity tells you how fast and in what direction it's moving, making it a full vector.

Magnitude and Direction

Magnitude and direction are key components distinguishing vectors from scalars. While magnitude concerns the size or length of a vector, direction defines its orientation in space.

• Magnitude: This is the numerical part of a vector. It shows how intense a quantity is. For instance, in the car example, the magnitude is 72 mph.
• Direction: This part tells you where the vector is pointing. In the car's case, it's 'due east', equivalent to 90 degrees from the north.

Together, magnitude and direction fully characterize vectors like velocity. This dual characteristic allows vectors to represent real-world situations more holistically by merging both how much and where aspects into one.
In summary, vectors like velocity differ from simple numbers, as they give a complete picture of motion or force by combining both magnitude and direction.