Vectors

Vectors are different from scalar numbers because they also include information about direction. Velocity, acceleration, force, and displacement are all vectors. Speed, time, and mass are all scalar quantities. Acceleration can be either a scalar or a vector, although in physics it’s usually considered a vector. For example, a car traveling at 45 mph is a speed, whereas a car traveling 45 mph NW is a vector. When you draw a vector, it’s an arrow that has a head and a tail, where the head points in the direction the force is pulling or the object is moving.

The coordinate system you use can be a compass (north, south, east and west) which is good for problems involving maps and geography, rectangular coordinates (x and y axes) which is good for most problems with objects traveling in two directions, or polar coordinates (radius and angle) which is good for objects that spin or rotate.

We have to get really good at vectors and modeling real world problems down on paper with them, because that’s how we’ll break things down to solve for our answers. If you’re already comfortable with vectors, feel free to skip ahead to the next lesson. If you find you need to brush up or practice a little more, this section is for you.

A resultant is the vector sum of all of the vectors, usually force vectors. You can’t just add the numbers (magnitudes) together! You have to account for the direction that you’re pushing the box in. Here’s what you need to know about vector diagrams and how to add vectors together:

A vector in two dimensions has components in both directions. Here’s how to add vectors together to get a single resultant vector using component addition as well as trigonometry (the law of cosines and the law of sines):

Vectors can be added together using the Pythagorean theorem if they are at right angles with each other (which components always are). Here’s more practice is how to do both rectangular and polar coordinate system components of a vector:

deals with problems where one object moves with respect to another. For example, an airplane might be traveling at 300 knots according to its airspeed indicator, but since it has a 20 knot headwind, the speed you see the airplane traveling at is actually 280 knots. You’ve seen this in action if you’ve ever noticed a bird flapping its wings but not moving forward on a really windy day. In that case, the velocity of the wind is equal and opposite to the bird’s velocity, so it looks like the bird’s not moving.

But what if the airplane encounters a crosswind? Something that’s not straight-on light a head or tail wind? Here’s how you break it down with vectors:

These types of problems aren’t limited to airplanes, though. Have you ever gone in a boat and drifted off course? Here’s what was happening from a physics point of view:

These types of boat problems usually ask for the following information to be calculated: what is the resultant velocity of the boat, how much time does it take to cross the river, and what distance does the boat drift off course due to the wind? Let’s practice this type of problem again so you really can get the hang of it.