You will determine the magnitude of vectors by finding the length of their representative directed line segments.
After completing this tutorial, you will be able to complete the following:
Recall that a quantity is a measurable attribute of an object. The size of a quantity is known as its magnitude. Consider a car traveling north for two hours at a speed of 50 miles per hour. The time the car spent traveling is two hours, the speed the car traveled is 50 miles per hour, and the displacement, or the length of the shortest path from the starting point to ending point, of the car is 100 miles. Here, the time, speed, and displacement of the car are all scalars, quantities consisting only of magnitude.
Other quantities, called vectors, consist of a magnitude and a direction. In the example of the moving car above, we know that it is moving north at a speed of 50 miles per hour. This is a vector quantity that consists of a magnitude (its speed is 50 miles per hour) and a direction (north), and is called the velocity of the car. Using the same example, we can see that the car finishes its journey at a point 100 miles north of its starting point. The displacement vector of the car's journey describes not only the length of the shortest path from the starting point of the car to the ending point, but also the direction from the starting point to the ending point.
You should note that vectors do not have an initial or a terminal point, only a direction and a magnitude. However, we can represent vectors with directed line segments that do have initial and terminal points. Thus, different directed line segments can represent the same vector.
We can also represent vectors in the plane with coordinates. The horizontal (or x-) coordinate of a vector is the horizontal distance from the initial point to the terminal point of any directed line segment representing the vector. Similarly, the vertical (or y-) coordinate of a vector is the vertical distance from the initial point to the terminal point of any directed line segment representing the vector. The vector's coordinates indicate the direction and magnitude of the vector. The vector with coordinates (2, 3) is represented by any directed line segment that has its terminal point two units to the right and three units up from its initial point. Notice that when we place the initial point of the directed line segment representing v= (2,3) at the origin, the vector's coordinates are exactly the coordinates of the terminal point:
Note that vector u also has vector coordinates (2, 3) since it is represented by a directed line segment that has its terminal point two units to the right and three units up from its initial point. However, the directed line segment representing u has initial point (-5, -2), and terminal point (-3, 1).
|Approximate Time||20 Minutes|
|Pre-requisite Concepts||Students should be able to define a vector as having a magnitude and direction, define a directed line segment, find the coordinates of a vector when given a representative directed line segment, represent vectors with directed line segments, and use the distance formula.|
|Type of Tutorial||Skills Application|
|Key Vocabulary||directed line segment, magnitude, magnitude of a vector|