What’s a Vector?
What’s a Vector?
A vector is a mathematical object possessing, and fully described by, a magnitude and a direction. It’s possible to talk about vectors simply in terms of numbers, but it’s often a lot easier to represent them graphically as arrows. The vector’s magnitude is equal to the length of the arrow, and its direction corresponds to where the arrow is pointing. Physicists commonly refer to the point of a vector as its tip and the base as its tail.
There are a number of ways to label vectors. You may have seen vectors labeled or A. This book will follow the convention you’ll find on SAT II Physics: vectors are written in boldface and vector magnitudes in plain script. For example, vector A has magnitude A.
Vectors vs. Scalars
In contrast to a vector quantity, a scalar quantity does not have a direction; it is fully described by just a magnitude. Examples of scalar quantities include the number of words in this sentence and the mass of the Hubble Space Telescope. Vector quantities you’ll likely come across quite frequently in physics include displacement, s; velocity, v; acceleration, a; force, F; momentum, p; electric field, E; and magnetic field, B.
When in doubt, ask yourself if a certain quantity comes with a direction. If it does, it’s a vector. If it doesn’t, it’s a scalar.
Example
Which of the following sentences deal with vector quantities?
  I. “I used to drive a 10-ton truck.”
 II. “You’ll find a gas station if you follow this road 20 miles due north.”
III. “The 10-volt battery is the one on your left.”
(A) I only
(B) II only
(C) III only
(D) II and III
(E) I, II, and III
“I used to drive a 10-ton truck” deals with mass, which is a scalar quantity. When we know that a truck weighs 10 tons, we don’t need to ask, “in what direction?” “You’ll find a gas station if you follow this road 20 miles due north” deals with the vector quantity of displacement. When asking directions to a gas station, you don’t simply want to know how far it is from where you are, but also in what direction you need to go. “The 10-volt battery is the one on your left” is slightly tricky: volts are a scalar quantity—you don’t ask in what direction the battery’s volts are going. However, you might be deceived by the mention of “on your left.” However, “on your left” is a reference to the battery, not to the volts. The magnitude “10 volts” doesn’t have a direction, so that quantity is a scalar. The answer is B.
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