3.1 Scalars and Vectors

A quantity that is fully described by a single number (with units) is called a scalar. Mass, temperature, volume, and energy are all scalars. We will often use an algebraic symbol to represent a scalar quantity. Thus m will represent mass, T temperature, V volume, E energy, and so on.

Our universe has three dimensions, so some quantities also need a direction for a full description. If you ask someone for directions to the post office, the reply “Go three blocks” will not be very helpful. A full description might be, “Go three blocks south.” A quantity having both a size and a direction is called a vector.

The mathematical term for the length, or size, of a vector is magnitude, so we can also say that a vector is a quantity having a magnitude and a direction.

FIGURE 3.1 shows that the geometric representation of a vector is an arrow, with the tail of the arrow (not its tip!) placed at the point where the measurement is made. An arrow makes a natural representation of a vector because it inherently has both a length and a direction. As you’ve already seen, we label vectors by drawing a small arrow over the letter that represents the vector: r⃗ for position, v ⃗ for velocity, a⃗ for acceleration.

FIGURE 3.1The velocity vector v ⃗ has both a magnitude and a direction.

The figure illustrates the characteristics used in a vector quantity.
FIGURE 3.1 Full Alternative Text

NOTE   Although the vector arrow is drawn across the page, from its tail to its tip, this does not indicate that the vector “stretches” across this distance. Instead, the vector arrow tells us the value of the vector quantity only at the one point where the tail of the vector is placed.

The magnitude of a vector can be written using absolute value signs or, more frequently, as the letter without the arrow. For example, the magnitude of the velocity vector in FIGURE 3.1 is v=|v ⃗|=5 m/s. This is the object’s speed. The magnitude of the acceleration vector a⃗ is written a. The magnitude of a vector is a scalar. Note that magnitude of a vector cannot be a negative number; it must be positive or zero, with appropriate units.

It is important to get in the habit of using the arrow symbol for vectors. If you omit the vector arrow from the velocity vector v ⃗ and write only v, then you’re referring only to the object’s speed, not its velocity. The symbols r⃗ and r, or v ⃗ and v, do not represent the same thing.