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Introducing Vectors on the Cartesian Coordinate Plane

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Introducing Vectors on the Cartesian Coordinate Plane

Algebra-2

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You will define vectors, scalars, equal vectors, opposite vectors, and zero vectors.

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Now You Know

After completing this tutorial, you will be able to complete the following:

  • Define directed line segments on the coordinate plane.
  • Define vectors as a quantity consisting of both direction and magnitude.
  • Represent vectors in the coordinate plane with either directed line segments or Cartesian coordinates.
  • Find the vector corresponding to a given directed line segment.
  • Determine if two vectors in the plane are equal.
  • Define the zero vector as a vector with length zero.
  • Define the opposite vector.

Everything You'll Have Covered

A quantity is a measurable attribute of an object. Often, when we measure something, we write down that measurement followed by a unit. For example, we say that the length of a table is 5 feet, or put more pedantically, the quantity of the length of the table has a size of 5 feet. The size of a quantity is known as its magnitude.

Some quantities consist solely of 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.

Informally, we can think of representing the velocity of the car with an arrow that points north, the direction it is moving. The length of the arrow represents the car's speed. If the car speeds up, we lengthen the arrow. If the car changes direction (say to move east at 70 miles per hour), we can introduce a new, longer arrow with the new direction:

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.

Vector Coordinates:

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).

Opposite Vectors:

Again, consider the example of the car that travels north at 50 miles per hour for two hours before stopping. Suppose it then turns around and heads back to its starting point, traveling south now at a speed of 75 miles per hour for an hour and 20 minutes. As discussed earlier, the car's displacement vector for the first part of the trip has a magnitude (100 miles) and a direction (north). We might write the coordinates of the vector as (0, 100). Notice that, although the car is traveling at a different speed on its return trip from its departing trip, the displacement of the car is simply the distance it traveled: 100 miles. The displacement vector of the car's return trip has a magnitude (100 miles) and a direction (south), and it has coordinates (0, -100). Vectors such as these two that have the same magnitude but opposite directions are known as opposite vectors.

Tutorial Details

Approximate Time 20 Minutes
Pre-requisite Concepts Students should know how to read and plot points in the Cartesian coordinate plane, and understand the concepts of line segment, point, ray, length, direction, and quantity.
Course Algebra-2
Type of Tutorial Concept Development
Key Vocabulary coordinates, directed line segment, scalars