Vectors
An understanding of vectors is essential for an understanding of physics. A vector is a quantity that has two aspects. It has a size, or magnitude, and a direction. In contrast, there a quantities called scalars that have only size.
Vectors are usually drawn as arrows. In the picture above, a two dimensional vector is drawn in yellow. This vector has two parts, or components. Its x-component, drawn in red, is positioned as if it were a shadow on the x-axis of the yellow vector. The white vector, positioned as a shadow on the y-axis, is the y-component of the yellow vector.
The word components, in the following context, means parts. So, to talk about the components of a vector, we mean the parts of a vector. For a great amount of situations the important parts of a vector are it's x-part and its y-part, or its x-component and its y-component. Here we will see how to find the x-component and the y-component of a vector.
The vector we will use in the following discussion is a force vector. The methods shown here, though, are true for any vector, such as a displacement, velocity, or acceleration vector.
Here on the (x, y) set of axes is the force vector that we will be dealing with:
If you drop a line from the tip of the original vector straight down to the x-axis and draw a vector along the x-axis from the origin to where this line hits the x-axis, then this newly drawn vector is the x-component of the original vector. In the diagram the line that was dropped down is shown as a thin black line and the x-component is shown as a red vector:
The tip of the x-component vector is directly below the tip of the original vector.
In the following diagram the thin black horizontal line marks how high up the original vector rises. A vertical vector, (parallel to the y-axis), which rises to this height is called the y-component of the original vector. The y-component is shown below in green:
When both the x-component and the y-component are drawn, a right triangle is formed with the original vector being the hypotenuse:
This right triangle will allow us to to do right triangle trigonometry using SOH-CAH-TOA definitions. Those definitions are briefly explained here, and they are explained in much more depth in the Right Triangle Trigonometry section in the Trigonometry Realms of Zona Land. Here we will use the Greek letter theta to represent an acute angle in a right triangle. The letter theta looks like this:
Here are the right triangle definitions for the sine, cosine, and tangent of an acute angle in a right triangle:
Usually we just summarize these three definitions with these three short sentences:
The sine equals opposite over hypotenuse.
The cosine equals adjacent over hypotenuse.
The tangent equals opposite over adjacent.
Again, if these definitions are new material, see the Right Triangle Trigonometry section for more information.
Now, let's get back to our right triangle formed by the original vector and its x-component and y-component. Here's the diagram, fully labeled:
Notice that the x-component forms the side adjacent to the 35 degree angle, and that the y-component forms the opposite side to the 35 degree angle.
Let's find the size of the x-component; that is, let's find the size of the adjacent side.
We know the hypotenuse, (316 Newtons), and we know the angle, (35 degrees). We want to find the length of the adjacent side, (x-component). What trigonometry function relates the hypotenuse, an acute angle and its adjacent side in a right triangle? The cosine function does. The math looks this way:
Now, since the original vector is named F, its x-component is named Fx. This would be read 'F sub x'. So, in the above math we should remove 'x-component' and replace that term with Fx, as in:
We can solve for Fx by doing a little algebra and looking up the cosine of thirty-five degrees:
So, the x-component of the original vector is equal in size to 259 Newtons.
Now, realize this: The method for finding the x-component described here will not tell you the sign, (+ or -) for its value. This method will only tell you the size of the component. Notice that the x-component is pointing to the right. This makes it a positive x-component. (It would be negative if it pointed to the left.) So, we would would finally conclude that the x-component has a size of positive 259 Newtons.
Okay, now let's find the size of the y-component; that is, let's find the size of the opposite side.
Again, we know the hypotenuse, (316 Newtons), and we know the angle, (35 degrees). We want to find the length of the opposite side, (y-component). It is the sine function that relates the hypotenuse, an acute angle and its opposite side in a right triangle. The math looks this way:
Now, since the original vector is named F, its y-component is named Fy which would be read 'F sub y'. So, in the above math we should remove 'y-component' and replace that term with Fy, as in:
We can solve for Fy much like we solved for Fx:
So, the y-component of the original vector is equal in size to 181 Newtons.
Again, like in the case of the x-component, we must look at the diagram to correctly interpret the sign, (+ or -), of this y-component. This y-component is pointing up, so it is positive. (It would be negative if it pointed down.) Finally, we conclude that the y-component has a size of positive 181 Newtons.
Vector Component
Addition Example
In this example we will be adding the two vectors shown below using the component method. The vectors we will be adding are displacement vectors, but the method is the same with any other type of vectors, such as velocity, acceleration, or force vectors.
The component method of addition can be summarized this way:
- Using trigonometry, find the x-component and the y-component for each vector. Refer to a diagram of each vector to correctly reason the sign, (+ or -), for each component.
- Add up both x-components, (one from each vector), to get the x-component of the total.
- Add up both y-components, (one from each vector), to get the y-component of the total.
- Add the x-component of the total to the y-component of the total then use the Pythagorean theorem and trigonometry to get the size and direction of the total.
Let's take this all one step at a time. First, let's visualize the x-component and the y-component of d1. Here is that diagram showing the x-component in red and the y-component in green:
The two components along with the original vector form a right triangle. Therefore, we can use right triangle trigonometry to find the lengths of the two components. That is, we can use the 'SOH-CAH-TOA' type of definitions for the sine, cosine, and tangent trigonometry functions.
Let's find the x-component of d1. Notice that the x-component is adjacent to the angle of 34 degrees, so, we will use the cosine function since it relates an acute angle, the adjacent side to that angle, and the hypotenuse of a right triangle:
Now, using trigonometry like this will not tell us the sign, (+ or -), of this component, (or any other). So, we must check the diagram for positive or negative directions. This x-component is aimed to the right, so, it is positive:
Now, let's find the y-component of d1. Notice that the y-component is opposite to the angle of 34 degrees, so, we will use the sine function since it relates an acute angle, the opposite side to that angle, and the hypotenuse of a right triangle:
Again, check the diagram for positive or negative directions. The y-component aims up, so, it is positive:
Here is the diagram now showing the values for the x-component and y-component of d1:
Next, let's see the x-component and the y-component of d2. Here is that diagram showing the x-component in red and the y-component in green:
We will now find the x-component of d2. Here the x-component is opposite the angle of 64 degrees, so we will use the sine function to find it:
Check the diagram for positive or negative directions. This x-component points to the left, so, it is negative:
And for our last component we will find the y-component of d2. The x-component is adjacent to the angle of 64 degrees in this diagram, so we will use the cosine function to find it:
Check the diagram for positive or negative directions. This y-component is aimed up, so, it is positive:
Here is the diagram showing our newly calculated values for the components of d2:
Now, we must add up like components to get the components for the total displacement.
To get the total x-displacement, add up all of the separate x-components:
To get the total y-displacement, add up all of the separate y-components:
So, when these two vectors, d1 and d2, are added, the total, or sum, has an x-component of 9.2 meters and a y-component of 30.1 meters.
To get the actual 2-D total displacement, add the total x-displacement and the total y-displacement. Here is a diagram with the total x-component shown in red and the total y-component shown in green and the 2_D total shown in blue:
Use the Pythagorean theorem to get the magnitude (size) of the total 2-D displacement:
Use the arctangent function to get the angle:
Check the diagram for NSEW notation:
Therefore, our final result for the total 2-D displacement can be stated as:
Here's a diagram that shows this result:
Again, the component method of addition can be summarized this way:
- Using trigonometry, find the x-component and the y-component for each vector. Refer to a diagram of each vector to correctly reason the sign, (+ or -), for each component.
- Add up both x-components, (one from each vector), to get the x-component of the total.
- Add up both y-components, (one from each vector), to get the y-component of the total.
- Add the x-component of the total to the y-component of the total then use the Pythagorean theorem and trigonometry to get the size and direction of the total.
