In the previous tutorial, we had defined some terminology; one instance being the difference between a scalar and a vector, where a scalar only has a magnitude (ie, mass, density, length, speed), and vectors have magnitude and direction (ie, displacement, velocity, and acceleration).
Now we will be talking about vectors and some properties of vectors. First off, for a vector, they have two components, magnitude which for a vector $ \vec{A} $, magnitude of $ \vec{A} $ is $ |\vec{A}| $, and direction which is given by $ \theta $. The magnitude of a vector can be thought of as the length of the vector and theta lets us know in what direction that length is being directed.
Image of a vector with magnitude A and direction denoted by theta.
Next we will talk about some operations with vectors. The first up being the addition of two vectors. Lets say we have two vectors $ \vec{A} $ and $ \vec{B} $, then the sum of these vectors would be $ \vec{A} + \vec{B} = \vec{R} $, where $ \vec{R} $ is the resultant vector, or sum of the two. Now, there are two different methods to this: the triangle method and the parallelogram method. Let's first look at the parallelogram method.
Depiction of vector addition to find the resultant vector, R, using the parallelogram method.
You may be wondering, how do you know which direction to place the vectors, and that is a good question. I like to start out with a "center of origin" or point on the page where I call (0,0), or center. From there, I draw my first vector. Then I start from the same origin point and draw my second vector. Now we have two sides to our parallelogram drawn, we can then do something similar in the image above where we take the two vectors $ \vec{A} $ and $ \vec{B} $ and make the two remaining sides of the parallelogram by redrawing the two vectors and connecting the ends. Then the resultant vector $ \vec{R} $ is simply the diagonal of the parallelogram, starting one end at your defined origin.
The next method is the triangle method. This is a little bit easier to see at first, but by practicing both methods, each will come naturally. Let's see a picture of the triangle method using the same $ \vec{A} $, $ \vec{B} $, and $ \vec{R} $ vectors as before.
Depiction of vector addition to find R using the triangle method.
WIth the triangle method, we start out by placing our vector $ \vec{B} $ from the origin and then connecting vector $ \vec{A} $ from the end of $ \vec{B} $. We then draw a vector starting from the start of $ \vec{B} $ to the end of $ \vec{A} $ and that is our resultant vector $ \vec{R} $
The next vector operation we will learn about is subtracting vectors. You can imagine, this is very similar to vector addition with the parallelogram and triangle methods, but the subtracted vector is rotated by 180°! Let's take a look at this using the triangle method from before.
Image of vector subtraction using the triangle method.
Now that we better understand vectors and how to use some operations on them, let us talk about Cartesian components for vectors. A little background, Cartesian coordinates is the typical coordinate system that we use (ie, x, y, and z for 3D spaces).
Imagine we have a force, $ \vec{F} $, and it points at some angle $ \theta $ from the x-axis. How can we describe the amount of that force that is in the x-direction or the y-direction? Let's say we didn't have the value for $ \theta $; how could we find it? Let's keep going to find out. :)
A force vector F drawn at an angle theta above the x-axis.
We can describe the components of $ \vec{F} $ by the following:
[ 1 ]
$$ |\vec{F_x}| = |\vec{F}| \cos{\theta} $$
[ 2 ]
$$ |\vec{F_y}| = |\vec{F}| \sin{\theta} $$
[ 3 ]
$$ |\vec{F}| = \sqrt{|\vec{F_x}|^2 + |\vec{F_y}|^2} $$
[ 4 ]
$$ \theta = \tan^{-1}{\frac{|\vec{F_y}|}{|\vec{F_x}|}} $$
Eq. (1) means that in order to get the magnitude, or length of $ \vec{F_x} $, we need to multiply the magnitude of the overall force vector by the cosine of the angle $ \theta $. This is because we are trying to find the "amount" of our force that is in the x-direction and the "amount" of our force that is in the y-direction. The same goes for Eq. (2) with $ \vec{F_y} $, except we use sin of our angle $ \theta $.
For Eq. (3) and Eq. (4), we can use trigonometry to find a relation between $ \vec{F} $ and its components: $ \vec{F_x} $ and $ \vec{F_y} $. We can also use trig to find the angle $ \theta $ as well. These equations are very very useful and we will use these plenty throughout the guide!