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Vectors and Modeling Situations with Vectors

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Vector Concepts

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Vectors and Modeling Situations with Vectors


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

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Now You Know

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

  • Know the definition of vectors as quantities that consist of a magnitude and direction.
  • Determine equivalent vectors.
  • Determine the opposite of a vector.
  • Determine the zero vector.
  • Represent vectors using coordinates or directed line segments.
  • Find the vector corresponding to a given directed line segment.
  • Use vectors to model situations in the plane.

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.

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, vector