\(\def \u#1{\,\mathrm{#1}}\) \(\def \abs#1{\left|#1\right|}\) \(\def \ast{*}\) \(\def \deg{^{\circ}}\) \(\def \ten#1{\times 10^{#1}}\) \(\def \redcancel#1{{\color{red}\cancel{#1}}}\) \(\def \BLUE#1{{\color{blue} #1}}\) \(\def \RED#1{{\color{red} #1}}\) \(\def \PURPLE#1{{\color{purple} #1}}\) \(\def \th#1,#2{#1,\!#2}\) \(\def \lshift#1#2{\underset{\Leftarrow\atop{#2}}#1}}\) \(\def \rshift#1#2{\underset{\Rightarrow\atop{#2}}#1}}\) \(\def \dotspot{{\color{lightgray}{\circ}}}\)

Vectors

Vectors are an important mathematical tool in physics, so we should take a break and talk more about what they are.

A scalar is a quantity that can be specified with just a number (and maybe a unit).

• The temperature is 25°C today.
• It took me 12 minutes to read that article.
• I drove 30 miles today.

A vector, on the other hand, is a quantity that involves a number (called the magnitude) and a direction.

• The nearest gas station is five miles east of here.
• The car is driving at 45 km/h southwest.
• I apply an upward force of 50N on the block.

There are many types of vector quantities in physics: forces, displacement, velocity, acceleration, and others as well. When we write the variable of a vector we often draw a little vector cap on top of it to indicate that it is a vector, like this: \(\vec F\) . On a diagram, we represent a vector by an arrow with a number or variable written next to it giving its magnitude. The magnitude of a vector is formally written as \(\abs{\vec F}\), although it's also common to just write F.

When we draw a vector on a diagram, we draw an arrow to indicate its direction and write the magnitude next to it, as shown here. In some situations it is important to have the length of the arrow be proportional to the magnitude, so here the 8N arrow is twice as long as the 4N arrow. In many circumstances, however, this isn't strictly necessary.