Give examples of three real-world quantities that could be represented by a vector and three quantities that could be represented by a single real number. Explain why some quantities can be represented by a number and others must be represented by a vector.

One of the most common uses vectors have in the real world is in measuring velocity. For example, in a very simplified case, we could say a car travels at some speed in the northern direction, and at some speed in the eastern direction. If the car is travelling at 50 mph north and 22 mph east, we could represent that fact with the velocity vector (50, 22), where the first component is the car's northern velocity and the second is its eastern velocity.

We could also represent the car's position and acceleration with vectors in the same way, breaking them up into northern and eastern components.

Three examples that we can (and which you probably do!) represent with real numbers are height (ex: 5.5 feet), weight (ex: 130 lbs), and age (ex: 23 years). They only require a single quantity, so a single number is the best way of representing them.