We saw how you can use trigonometry to find the components of a vector. If a car’s velocity vector is 45 mph at a 30 degree angle to the x-axis, you can use the cosine of 30 to figure out it’s moving 39 mph in the x-direction. Using the sine of 30 you can find it’s going 22.5 mph in the y-direction.

A way to write this mathematically is with i and j notation. The letters i and j are unit vectors in the x and y direction respectively. The x component is multiplied times i and the y component is multiplied times j. Our vector for a car moving at 45 mph at a 30 degree angle is formally written as v = 39i + 22.5j and you can include the mph (units) if you want to be super correct.

Notice when we drew the velocity vector for our Smart Car, we also put in two smaller arrows. These arrows point exclusively in the x and y directions, and are the x and y components of the velocity vector. These components make a triangle with the velocity vector, which is the hypotenuse.

The mathematical study of triangles is called trigonometry, and one thing trigonometry is useful for is working with vectors. Even if you don’t know trigonometry, you’ve probably messed with the SIN, COS, and TAN buttons on a calculator. We can use the definition of a cosine (COS) to find the length of the x-component. On a calculator, type in 30, hit COS, and multiply that number by the hypotenuse (45 mph). You find that in the x direction only, the car is moving 39 mph.

Let’s talk about what a vector is. Vectors are often important in science and engineering for describing real-world problems in the language of mathematics. Say you’ve got a nice Smart Car, and you want to tell someone how fast you’re driving it. Speed is a scalar quantity, consisting of a number and some units. You can give your speed as 45 mph. (Or even better 20.1 m/s which we like because it’s in SI units!)

But what if you wanted to give your velocity? Velocity is different than speed, because it’s a vector quantity. This means the direction is also important. For example, check out the lower Smart Car, which is going 45 mph at a 30-degree angle to the x-axis. This is a velocity, because we’ve given the speed (magnitude) and the direction.