Beverly has taught mathematics at the high school level and has a doctorate in teaching and learning.
This lesson explores vectors, operations with vectors, and modern uses of vectors. Using relevant examples and diagrams, the lesson will demonstrate the applications of vectors in the world.
What Is a Vector?
A vector represents a quantity that has both magnitude (distance) and direction. For example, when you travel 16 kilometers south, your journey may be represented as a vector quantity. We know that you are traveling a distance of 16 km and we know that you are heading south. Force and velocity are some examples of vector quantities.
Scalar quantities only have magnitude, and are used with vector quantities. If you are told that Sam's car is traveling at 65 miles per hour, the only information you are told is the scalar measurement speed. However, when you hear that Sam's car is traveling southwest at 65 miles per hour, you are given a vector measurement. This is true because you now know both magnitude and direction. A vector is important in physics; it is important in aeronautics, space, and travel in general. Pilots and sailors use vector quantities to get to and from their destinations safely.
The method of representing vectors is very important to the understanding of vectors. We use a ray to represent a vector. We name a vector using lowercase or uppercase letters. Look at this image. Let's talk about a few important things:
1) Take note that vector a is represented by both uppercase and lower case letters.
2) When a vector is represented by upper case letters, we use this notation. No matter the direction of the vector, it is always written in this form. We name the vectors from tail to head (or arrowhead). In our example the vector is AB not BA. A is the tail, B is the head.
3) Vector b is only represented by a lowercase letter. Let us assume that vector b is opposite in direction from vector a. If this is true, then vector b is called an opposite vector, which means the vectors have the same magnitude but different direction.
4) Since vector b is opposite to vector a, then vector b may also be written as -a. This means b = -a. We negate a vector to show that it is has the same magnitude as another vector that is heading in an opposite direction. It's like two parallel streets, one heading north and one south.
Operations with Vectors
We can add and subtract vectors. We can add vectors by connecting head to tail. When we add two vectors, the final vector is called the resultant and is noted by a lowercase r. Let us take a look at this image:
In this diagram, we have three vectors. We added vector q to vector p. Our resultant vector is r. Our starting point is at the tail of vector q. Our destination is to get to the head of vector p. Of course, instead of going from the yellow vector to the blue vector, we could easily travel straight on the vector r. Vectors help us to see direction more realistically. If you were traveling along this route, it definitely would make sense to travel on vector r to get to where you are going faster. However, this is not always the case.
By adding the vectors we get q + p = r, which is the same as saying p + q = r. However, our diagram would be a bit different because then the blue vector would have to be first.
Notice that we didn't change the direction of any vector. However, the layout of our diagram changes because of our starting point. That is why it is important to label and draw vectors accordingly. This is also why vectors are very important to pilots, because where you are starting from depends on the journey you take and where it leads you.
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A displacement vector is also very important in our discussion of vectors and refers to the final position of a vector after a series of movements. Zero displacement can be explained with this simple example. What happens if you take four steps forward and four steps backward? You would end up at your starting place, right? That is zero displacement; you made no progress. Your final position is where you started from, so in a sense, although you moved, you made no progress.
Let us look at an example to calculate the displacement:
Let us pretend that a sailor is starting out at vector a to get to vector d. Vector r represents the resulting displacement. Therefore, we would add a + b + c + d = r. Notice that vectors b and d are the same in magnitude but different direction. Therefore, we could also represent the sum as a + b + c - b = r, since d = - b. Our displacement then is a + c = r.
It would seem ridiculous to travel forward c units to go backward c units; because of unavoidable forces of nature, a pilot or sailor may have to take these trips. However, in most cases they take the shortest possible route.
We can multiply vectors by scalar quantities. For example, vector a can be doubled, which gives us 2a. This means the length of vector a is doubled, but the direction is unchanged.
Another important notation of vectors is that the magnitude of a vector is represented using the absolute value sign. That is the magnitude or distance of a vector a is written like this:
If x = 10 and y = 15, then the magnitude of vector a would be the square root of (100 + 225). This gives us 18.02 units. In other words, if you go 10 units south and 15 units west, how far did you really travel in the southwest direction? It would be 18.02 units SW.
A vector is a quantity that has both magnitude and direction, such as a velocity of 60 miles per hour, heading north.
A scalar quantity only has magnitude, such as time: 8 days.
We can add vectors by connecting head to tail.
Two vectors that have the same magnitude but head in opposite directions are opposite vectors. A negative sign is used to represent opposite direction, such as vector a and -a.
We can perform operations on vectors such as adding, subtracting, and multiplying.
A displacement vector is the resulting vector after a series of movements.
Zero displacement means that a movement occurred the same number of units in opposite directions (for example, 10 units north and 10 units south), such that the object is back at its original position.
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