Vectors

What is a vector?

In our Physics course you can think of a vector as a measurement that requires 2 parts to fully record it. The two parts we will be dealing with are magnitude (an amount) and direction.

The other type of measurement we will be using is a scalar. Scalars only need a magnitude to be fully represented. Examples include time (5 sec), mass (8 kg), and height (2 m).

Why do they matter?

Vector quantities are often represented with an arrow at this level in Physics. The values can be added and subtracted through a simple graphical method or by following some mathematical routines. When you add vectors, though, the direction needs to be considered as well. For example, displacement is a measurement of how a final position differs from the original position. If a person moves 3 meters and then 4 meters how far away is she from where she started?

In this case you can see that the direction matters because she could be any distance between 1 meter away (ex. 1) to 7 meters away (ex. 2) to somewhere in between (5 meters away in example 3, but at different angles any value between 1 and 7 meters is possible).

Tip to tail addition of vectors

This example above shows how simple it is to add and subtract one dimensional vectors using what is called the tip to tail method of adding vectors. In this method you simply draw the first vector as an arrow, then draw the second one from the end point of the first arrow (put the tail of the new vector at the tip of the 1st one). The result of vector addition is called a resultant and is found by drawing a new arrow from the origin of the first vector to the tip of the last vector added (this works for adding more than just 2 vectors). In the case of example A above, the resultant is a vector of magnitude 1m with a direction of -x (or to the left). Interestingly, it doesn't matter which order you draw these as you can see below, placing the 4 m left first, then coming 3 meters back to the right will produce the same resultant of 1 meter to the left.

When you have vectors act on more than one axis, you will likely need to use the pythagorean theorem to find the magnitude and trig functions to find the direction. You can simply memorize the formula for this until you get better at using them, though. For example C above, you find the magnitude of the resultant by recognizing this is a right triangle with sides of 3 and 4 meters. The resultant vector can be described as 5 m at an angle of 60.6º above the X axis.

Finding components of vectors

We chose simple vectors in the examples above, and really, most of the vectors used in AP Physics 1 are like this. However, there are times when we have an angled vector - most often an angled velocity or an angled force, where the easiest and fastest way to deal with that problem is to split this vector into its vector components. What this means, is split up the angled vector into two simpler vectors to see how it affects each axis separately. To do this, we will make a right triangle where the hypotenuse is the vector we started with. Then identify the opposite and adjacent sides of the triangle we formed and use the appropriate trig functions. For example, a ball launched at a velocity of 60 m/s at an angle of 30 degrees above the horizontal can be split using this method:

Practice

Practice adding these vectors and checking your work here: At the physicsclassroom.com

Video Lesson

I made this video several years ago. If you prefer to learn from a video, this might help you.