Lesson 11E Plane Geometry Practice Understanding

Learning Focus

Model real-world scenarios with vectors, matrices and trigonometry.

What kinds of real-world situations can be modeled with vectors and matrices?

Open Up the Math: Launch, Explore, Discuss

Jon’s father is a pilot, and he is using vector diagrams to explain some principles of flight to Jon. His father has drawn the following diagram to represent a plane that is being blown off course by a strong wind. The plane is heading northeast as represented by , and the wind is blowing toward the southeast as represented by .

1.

  1. Based on the diagram, what is the plane’s speed and what is the wind’s speed? (The vector diagram represents the speed of the plane in still air.) Note: Each unit on the grid is .

  2. Use the diagram to find the ground speed of the plane, which will result from a combination of the plane’s speed and the wind’s speed. Also, indicate on the diagram the direction of motion of the plane relative to the ground.

Graph with vector p, vector w with compose rose in the bottom right of graph x505050100100100150150150y–50–50–50505050100100100000

2.

How could you describe the actual direction of the plane’s motion using an angle?

While early compasses were marked to indicate the 32 wind directions named by sailors, modern compasses are marked using the degrees in a circle, with / representing North, as East, as South, and as West.

3.

How could you describe the direction of the plane’s motion using a compass heading?

4.

The wind changes direction during the flight so that it is blowing at the same speed as before, but at a compass heading. Find the speed and compass heading of the plane now.

5.

Write a matrix equation that will rotate the vector representing the motion of the plane that you found in problem 4 around the origin.

6.

Write a matrix equation that will reflect the vector representing the wind in problem 4 over the -axis.

7.

Find the actual ground speed and compass heading of a plane using the vector for the motion of the plane found in problem 5 and the wind vector found in problem 6.

Ready for More?

Describe a procedure for finding the compass heading for a vector , for vectors facing in each of the following directions:

a.

Up, and to the right

b.

Up, and to the left

c.

Down, and to the left

d.

Down, and to the right

Takeaways

When working with contexts that involve vectors, I can choose and translate among different forms for representing vectors: (represent the vector in the diagram in each of these different ways)

Directed line segment on a vector diagram (given)

Horizontal and vertical components:

Single-column matrix:

graph with vector <-5,3> with initial point at (0,0) x–5–5–5555y–5–5–5555000

Magnitude and angle of rotation

Magnitude:

Angle of rotation:

Magnitude and compass heading

Magnitude:

Compass heading:

Lesson Summary

In this lesson, we used vectors and matrices to represent a real-world context of a plane’s flight path being impacted by the wind. While representing this context, we had to represent vectors in a variety of ways, including directed line segments, horizontal and vertical components, single-column matrices, and magnitude and angle of direction.

Retrieval

1.

Describe the correlation of the data and sketch a trend line on the graph.

Graph of scatter plot x555101010y555101010000

2.

Write the equation for the trend line.

3.

Solve the system of equations by elimination.