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ZingPath: Vector Operations

Adding Vectors Geometrically and Algebraically

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Vector Operations

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Lesson Focus

Adding Vectors Geometrically and Algebraically


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You will be introduced to vector addition and, using real-life examples, add two vectors geometrically by using the parallelogram and head-to-tail methods, and algebraically by finding the resultant vector coordinate-wise.

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Now You Know

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

  • Add vectors using the parallelogram method.
  • Add vectors using head-to-tail method.
  • Add vectors coordinate-wise.
  • Explain the triangle inequality for vector addition.

Everything You'll Have Covered

Let be two vectors in the coordinate plane. Recall that vectors are quantities consisting of a direction and a magnitude. They are represented in the plane by directed line segments, and written algebraically using Cartesian coordinates.

Vectors are useful for modeling many types of quantities. For example, force consists of a direction and a magnitude, and is therefore a vector quantity.

To see how vectors are useful for modeling force, imagine two people pushing in different directions on a large box.

If the box were on a slippery surface and given sufficient force, it would move according to the joint effort of the pushers. In the situation depicted by the diagram above, the box would move diagonally across the room.

The combined forces is called the resultant force. The magnitude of this force should be measured in a unit such as Newtons, but we omit them in order to avoid confusion.

The resultant force can be determined by drawing a parallelogram in the diagram, as shown below. In this diagram, the resultant force is represented by the diagonal of the parallelogram.

This is an example of vector addition. Geometrically, we can visualize vector addition by using a parallelogram. Under this interpretation, the sum of two vectors is then represented along the diagonal of the parallelogram formed by representations of the vectors. The sum of two vectors, is denoted by and called the resultant vector.

We can also find a more useful algebraic rule for vector addition. In the diagram above, suppose . The dotted lines in the diagram represent the same vectors, , but with different directed line segments. By following along the solid and then the dotted line, we end up at point . This method is sometimes called the head-to-tail method.

Tutorial Details

Approximate Time 30 Minutes
Pre-requisite Concepts Students should know the definition of vector as a quantity consisting of both direction and magnitude; be able to represent vectors in the coordinate plane with either directed line segments or coordinates; and add, subtract, and multiply real numbers.
Course Algebra-2
Type of Tutorial Concept Development
Key Vocabulary adding vectors, head-to-tail method for adding vectors, parallelogram method for adding vectors