Lesson 12 The Arithmetic of Vectors Solidify Understanding
Jump Start
Notice and Wonder:
Examine the following list of quantities, and describe at least two things you notice and something you are wondering about:
A book costs
. A
wind blowing towards the northeast. on a bookshelf. The force of gravity accelerates objects at
toward the ground. A plane flying
directly west. of silver.
Learning Focus
Represent quantities that have both magnitude and direction using vectors, and examine the arithmetic of vectors.
How do we represent quantities that have both magnitude (size) and direction?
How do we add or subtract quantities that have both magnitude and direction?
Open Up the Math: Launch, Explore, Discuss
The following diagram shows a triangle that has been translated to a new location, and then translated again. Arrows have been used to indicate the movement of one of the vertex points through each translation. The result of the two translations can also be thought of as a single translation, as shown by the third arrow in the diagram.
1.
Draw arrows to show the movement of the other two vertices through the sequence of translations, and then draw an arrow to represent the resultant single translation. What do you notice about each set of arrows?
A vector is a quantity that has both magnitude and direction. The arrows you drew on the diagram represent translations as vectors—each translation has magnitude (the distance moved) and direction (the direction in which the object is moved). Arrows, or directed line segments, are one way of representing a vector.
Addition of Vectors
In the example provided, two vectors
2.
Study each of the following three methods for adding vectors, then try each method to add vectors
a.
Method 1: End-to-end
The diagram given in problem 1 illustrates the end-to-end strategy of adding two vectors to get a resultant vector that represents the sum of the two vectors. In this case, the resulting vector shows that a single translation could accomplish the same movement as the combined sum of the two individual translations, that is,
b.
Method 2: The parallelogram rule
Since we can relocate the arrow representing a vector, draw both vectors starting at a common point. Often both vectors are relocated so they have their tail ends at the origin. These arrows form two sides of a parallelogram. Draw the other two sides. The resulting sum is the vector represented by the arrow drawn from the common starting point (for example, the origin) to the opposite vertex of the parallelogram.
Question to think about: How can you determine where to put the missing vertex point of the parallelogram?
c.
Method 3: Using horizontal and vertical components
Each vector consists of a horizontal component and a vertical component. For example, vector
d.
Explain why each of these methods gives the same result.
Question to think about: How can the components of the individual vectors be combined to determine the horizontal and vertical components of the resulting vector
3.
Examine vector
Magnitude of Vectors
The symbol
Using the diagram given at the beginning of the task, devise a method for finding the magnitude of a vector, and use your method to find the following. Be prepared to describe your method for finding the magnitude of a vector.
4.
5.
6.
Scalar Multiples of Vectors
A vector can be stretched by multiplying the vector by a scale factor. For example,
Draw each of the following vectors on a coordinate grid:
7.
8.
9.
10.
Other Applications of Vectors
The concept of a vector has been illustrated using translation vectors in which the magnitude of the vector represents the distance a point gets translated. There are other quantities that have magnitude and direction, but the magnitude of the vector does not always represent length.
For example, a car traveling
11.
A swimmer is swimming across a river with a speed of
12.
Two teams are participating in a tug-of-war. One team exerts a combined force of
Ready for More?
List all of the quantities you can think of that involve both magnitude and direction. Then create a scenario based on one of your types of vector quantities, like the scenarios in problems 11 and 12. You might want to consult with the physics teacher at your school for examples.
Takeaways
There are three strategies I can use to add vectors:
To subtract vectors:
To multiply a vector by a scalar:
Adding Notation, Vocabulary, and Conventions
Scalar quantities:
Vector quantities:
Vocabulary
- directed distance
- direction of a vector
- magnitude of a vector
- scalar quantity
- vector, vector quantity
- Bold terms are new in this lesson.
Lesson Summary
In this lesson, we learned how to represent quantities that have both magnitude and direction, such as a wind blowing at
1.
Rotate point
counterclockwise about the origin, and label it . Rotate point
counterclockwise about the origin, and label it . Find the equation of the circle that passes through
, , and .
2.
Multiply the matrices.