Definition 1.1.1. Vectors.
A vector is a mathematical object with both a magnitude (a size or amount) and a direction.
Section 1.1 Introduction to Vectors
In diagrams, vectors are represented by arrows. The length of the arrow represents the magnitude of the vector and the arrowhead indicates the direction of the vector. Symbolically, you draw an arrow on top of a symbol to denote a vector quantity, such as \(\vec{A}\text{.}\) On its own, the symbol \(A\) would represent the magnitude of the vector, which is sometimes written \(|\vec{A}|\text{.}\)
Definition 1.1.2. Scalars.
A scalar is a mathematical object that has a value (which can in general be either positive or negative) but has not direction.
Suppose you have a vector \(\vec{A}\text{.}\) The negative vector \(-\vec{A}\) has the same magnitude as vector \(\vec{A}\text{,}\) but points in the opposite direction.
Exercises Activities
1. Explore Vectors and Scalars.
Make a list of physical quantities you are familiar with from your everyday experience. For each quantity, do you think it would be better to represent it as a scalar or as a vector?
Answer.
Position, velocity, acceleration, force, and momentum are examples of physical quantities that are represented by vectors. In contrast, time, mass, charge, and temperature are represented by scalars.
2. Explore Magnitude.
A vector is typically represented graphically by an arrow. Which property of the arrow do you think represents the magnitude of the vector? How would you represent two vectors that have the same direction but different magnitudes?
References References
[1]
Vectors vs. Scalars by Dr. Michelle Tomasik from MIT 8.01 Classical Mechanics, Fall 2016, used under Creative Commons BY-NC-SA.
