Eureka Math Precalculus Module 2 Lesson 5 Answer Key

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Eureka Math Precalculus Module 2 Lesson 5 Exercise Answer Key

Opening Exercise

Compute:
a. (-10+9i)+(7-5i)
Answer:
-3+4i

b. 5∙(2+3i)
Answer:
10+15i

c. \(\left(\begin{array}{c}
5 \\
-6
\end{array}\right)\)+\(\left(\begin{array}{c}
2 \\
7
\end{array}\right)\)
Answer:
\(\left(\begin{array}{c}
7 \\
1
\end{array}\right)\)

d. -2\(\left(\begin{array}{c}
3 \\
-3
\end{array}\right)\)
Answer:
\(\left(\begin{array}{c}
-6 \\
6
\end{array}\right)\)

Exercises

Exercise 1.
Let x=\(\left(\begin{array}{c}
5 \\
1
\end{array}\right)\), y=\(\left(\begin{array}{c}
2 \\
3
\end{array}\right)\). Compute z=x+y, and draw the associated parallelogram.
Eureka Math Precalculus Module 2 Lesson 5 Exercise Answer Key 16
Answer:
\(\left(\begin{array}{l}
7 \\
4
\end{array}\right)\)

Exercise 2.
Let x=\(\left(\begin{array}{c}
-4 \\
2
\end{array}\right)\), y=\(\left(\begin{array}{c}
1 \\
3
\end{array}\right)\). Compute z=x+y, and draw the associated parallelogram.
Eureka Math Precalculus Module 2 Lesson 5 Exercise Answer Key 17
Answer:
\(\left(\begin{array}{c}
-3 \\
5
\end{array}\right)\)

Exercise 3.
Let x=\(\left(\begin{array}{c}
3 \\
2
\end{array}\right)\), y=\(\left(\begin{array}{c}
-1 \\
-3
\end{array}\right)\). Compute z=x+y, and draw the associated parallelogram.
Eureka Math Precalculus Module 2 Lesson 5 Exercise Answer Key 18
Answer:
\(\left(\begin{array}{c}
2 \\
-1
\end{array}\right)\)

Exercise 4.
Let x=\(\left(\begin{array}{c}
3 \\
2
\end{array}\right)\). Compute z=2x, and plot x and z in the plane.
Eureka Math Precalculus Module 2 Lesson 5 Exercise Answer Key 19
Answer:
\(\left(\begin{array}{l}
6 \\
4
\end{array}\right)\)

Exercise 5.
Let x=\(\left(\begin{array}{l}
-6 \\
3
\end{array}\right)\). Compute z=\(\frac{1}{3}\)x, and plot x and z in the plane.
Eureka Math Precalculus Module 2 Lesson 5 Exercise Answer Key 20
Answer:
\(\left(\begin{array}{c}
2 \\
-1
\end{array}\right)\)

Exercise 6.
Let x=\(\left(\begin{array}{c}
1 \\
-1
\end{array}\right)\). Compute z=-3x, and plot x and z in the plane.
Eureka Math Precalculus Module 2 Lesson 5 Exercise Answer Key 21
Answer:
\(\left(\begin{array}{c}
-3 \\
3
\end{array}\right)\)

Exercise 7.
Let x=\(\left(\begin{array}{l}
3 \\
1 \\
1
\end{array}\right)\) and y=\(\left(\begin{array}{l}
1 \\
3 \\
1
\end{array}\right)\). Compute z=x+y, and then plot each of these three points.
Eureka Math Precalculus Module 2 Lesson 5 Exercise Answer Key 22
Answer:
\(\left(\begin{array}{l}
4 \\
4 \\
2
\end{array}\right)\)

Exercise 8.
Let x=\(\left(\begin{array}{l}
3 \\
0 \\
0
\end{array}\right)\) and y=\(\left(\begin{array}{l}
0 \\
3 \\
0
\end{array}\right)\). Compute z=x+y, and then plot each of these three points.
Eureka Math Precalculus Module 2 Lesson 5 Exercise Answer Key 21.2
Answer:
\(\left(\begin{array}{l}
3 \\
3 \\
0
\end{array}\right)\)

Exercise 9.
Let x=\(\left(\begin{array}{l}
1 \\
1 \\
1
\end{array}\right)\). Compute z=4x, and then plot each of the three points.
Eureka Math Precalculus Module 2 Lesson 5 Exercise Answer Key 26
Answer:
\(\left(\begin{array}{l}
4 \\
4 \\
4
\end{array}\right)\)

Exercise 10.
Let x=\(\left(\begin{array}{l}
2 \\
4 \\
4
\end{array}\right)\). Compute z=-\(\frac{1}{2}\)x, and then plot each of the three points. Describe what you see.
Eureka Math Precalculus Module 2 Lesson 5 Exercise Answer Key 27
Answer:
\(\left(\begin{array}{l}
-1 \\
-2 \\
-2
\end{array}\right)\)
In the resultant, the direction has reversed, and the length of the ray is half the original.

Eureka Math Precalculus Module 2 Lesson 5 Problem Set Answer Key

Question 1.
Find the sum of the following complex numbers, and graph them on the complex plane. Trace the parallelogram that is formed by those two complex numbers, the resultant, and the origin. Describe the geometric interpretation.
x=\(\left(\begin{array}{l}
2 \\
3
\end{array}\right)\), y=\(\left(\begin{array}{l}
3 \\
2
\end{array}\right)\)
Eureka Math Precalculus Module 2 Lesson 5 Problem Set Answer Key 33
Answer:
\(\left(\begin{array}{l}
5 \\
5
\end{array}\right)\) The two points, the resultant, and the origin form a parallelogram.

b. x=\(\left(\begin{array}{l}
2 \\
4
\end{array}\right)\), y=\(\left(\begin{array}{l}
-4 \\
2
\end{array}\right)\)
Eureka Math Precalculus Module 2 Lesson 5 Problem Set Answer Key 34
Answer:
\(\left(\begin{array}{l}
-2 \\
6
\end{array}\right)\) The two points, the resultant, and the origin form a parallelogram.

c. x=\(\left(\begin{array}{l}
2 \\
1
\end{array}\right)\), y=\(\left(\begin{array}{l}
-4 \\
-2
\end{array}\right)\)
Eureka Math Precalculus Module 2 Lesson 5 Problem Set Answer Key 35
Answer:
\(\left(\begin{array}{l}
-2 \\
-1
\end{array}\right)\) The resultant is double the magnitude of the original vector in the opposite direction.

d. x=\(\left(\begin{array}{l}
1 \\
2
\end{array}\right)\), y=\(\left(\begin{array}{l}
2 \\
4
\end{array}\right)\)
Eureka Math Precalculus Module 2 Lesson 5 Problem Set Answer Key 36
Answer:
\(\left(\begin{array}{l}
3 \\
6
\end{array}\right)\) The resultant is triple the length of the original vector in the same direction.

Question 2.
Simplify and graph the complex number and the resultant. Describe the geometric effect on the complex number.
a. x=\(\left(\begin{array}{l}
1 \\
2
\end{array}\right)\), k=2, kx= ?
Eureka Math Precalculus Module 2 Lesson 5 Problem Set Answer Key 37
Answer:
\(\left(\begin{array}{l}
2 \\
4
\end{array}\right)\) The point is dilated by a factor of 2.

b. x=\(\left(\begin{array}{l}
-6 \\
3
\end{array}\right)\), k=-\(\frac{1}{3}\), kx= ?
Eureka Math Precalculus Module 2 Lesson 5 Problem Set Answer Key 37.2
Answer:
\(\left(\begin{array}{l}
2 \\
-1
\end{array}\right)\) The point is dilated by a factor of \(\frac{1}{3}\)and mapped to the other side of the origin on the same line.

c. x=\(\left(\begin{array}{c}
3 \\
-2
\end{array}\right)\), k=0, kx= ?
Eureka Math Precalculus Module 2 Lesson 5 Problem Set Answer Key 40
Answer:
\(\left(\begin{array}{c}
0 \\
0
\end{array}\right)\) The point is mapped to the origin, (0,0).

Question 3.
Find the sum of the following points, graph the points and the resultant on a three-dimensional coordinate plane, and describe the geometric interpretation.
a. \(\left(\begin{array}{l}
1 \\
1 \\
2
\end{array}\right)\), y=\(\left(\begin{array}{l}
2 \\
1 \\
2
\end{array}\right)\)
Eureka Math Precalculus Module 2 Lesson 5 Problem Set Answer Key 41
Answer:
\(\left(\begin{array}{l}
3 \\
2 \\
4
\end{array}\right)\) The two points, the resultant, and the origin are on the same plane and form a parallelogram.

b. x=\(\left(\begin{array}{l}
1 \\
1 \\
1
\end{array}\right)\), y=\(\left(\begin{array}{l}
2 \\
2 \\
2
\end{array}\right)\)
Eureka Math Precalculus Module 2 Lesson 5 Problem Set Answer Key 42
Answer:
\(\left(\begin{array}{l}
3 \\
3 \\
3
\end{array}\right)\) Three points all collapse onto the same line in space.

c. x=\(\left(\begin{array}{l}
2 \\
0 \\
0
\end{array}\right)\), y=\(\left(\begin{array}{l}
0 \\
2 \\
0
\end{array}\right)\)
Eureka Math Precalculus Module 2 Lesson 5 Problem Set Answer Key 43
Answer:
\(\left(\begin{array}{l}
2 \\
2 \\
0
\end{array}\right)\) The two points, the resultant, and the origin are on the same plane and formed a parallelogram.

Question 4.
Simplify the following, graph the point and the resultant on a three-dimensional coordinate plane, and describe the geometric effect.
a. x=\(\left(\begin{array}{l}
2 \\
1 \\
1
\end{array}\right)\), k=2, kx= ?
Eureka Math Precalculus Module 2 Lesson 5 Problem Set Answer Key 44
Answer:
\(\left(\begin{array}{l}
4 \\
2 \\
2
\end{array}\right)\) The point is dilated by a factor of 2.

b. x=\(\left(\begin{array}{l}
2 \\
2 \\
2
\end{array}\right)\), k=-1, kx= ?
Eureka Math Precalculus Module 2 Lesson 5 Problem Set Answer Key 45
Answer:
\(\left(\begin{array}{c}
-2 \\
-2 \\
-2
\end{array}\right)\) The point is mapped to the other side of the origin on the same line.

Question 5.
Find:
a. Any two different points whose sum is \(\left(\begin{array}{l}
0 \\
0
\end{array}\right)\)
Answer:
Answers vary. \(\left(\begin{array}{l}
2 \\
2
\end{array}\right)\), \(\left(\begin{array}{l}
-2 \\
-2
\end{array}\right)\)

b. Any two different points in three dimensions whose sum is \(\left(\begin{array}{l}
0 \\
0 \\
0
\end{array}\right)\)
Answer:
Answers vary. \(\left(\begin{array}{l}
2 \\
2 \\
2
\end{array}\right)\), \(\left(\begin{array}{l}
-2 \\
-2 \\
-2
\end{array}\right)\)

c. Any two different complex numbers and their sum will create the degenerate parallelogram.
Answer:
Answers vary. \(\left(\begin{array}{l}
1 \\
2
\end{array}\right)\), \(\left(\begin{array}{l}
2 \\
4
\end{array}\right)\), as long as the answers are in the relation of \(\frac{y_{1}}{x_{1}}\) =\(\frac{y_{2}}{x_{2}}\)

d. Any two different points in three dimensions and their sum lie on the same line
Answer:
Answers vary. \(\left(\begin{array}{l}
1 \\
1 \\
1
\end{array}\right)\), \(\left(\begin{array}{l}
2 \\
2 \\
2
\end{array}\right)\)

e. A point that is mapped to \(\left(\begin{array}{c}
1 \\
-3
\end{array}\right)\) after multiplying –2
Answer:
\(\left(\begin{array}{c}
-\frac{1}{2} \\
\frac{3}{2}
\end{array}\right)\)

f. A point that is mapped to \(\left(\begin{array}{c}
\frac{1}{2} \\
-2 \\
4
\end{array}\right)\) after multiplying –\(\frac{2}{3}\)
Answer:
\(\left(\begin{array}{c}
-\frac{3}{4} \\
3 \\
-6
\end{array}\right)\)

Question 6.
Given x=\(\left(\begin{array}{l}
2 \\
1
\end{array}\right)\) and y=\(\left(\begin{array}{l}
-4 \\
-2
\end{array}\right)\):
a. Find x+y, and graph the parallelogram that is formed by x, y, x+y, and the origin.
Eureka Math Precalculus Module 2 Lesson 5 Problem Set Answer Key 52
Answer:
\(\left(\begin{array}{l}
-2 \\
-1
\end{array}\right)\)

b. Transform the unit square by multiplying it by the matrix \(\left(\begin{array}{ll}
2 & -4 \\
1 & -2
\end{array}\right)\), and graph the result.
Answer:
Eureka Math Precalculus Module 2 Lesson 5 Problem Set Answer Key 53

c. What did you find from parts (a) and (b)?
Answer:
They both have degenerate parallelograms.

d. What is the area of the parallelogram that is formed by part (a)?
Answer:
The area is 0.

e. What is the determinant of the matrix \(\left(\begin{array}{ll}
2 & -4 \\
1 & -2
\end{array}\right)\)?
Answer:
\(\left|\begin{array}{ll}
2 & -4 \\
1 & -2
\end{array}\right|\)=2(-2)-1(-4)=-4+4=0 The determinant is 0.

f. Based on observation, what can you say about the degenerate parallelograms in part (a) and part (b)?
Answer:
For two complex numbers, if \(\frac{\boldsymbol{b}_{\mathbf{1}}}{\boldsymbol{a}_{\mathbf{1}}}\) =\(\frac{\boldsymbol{b}_{2}}{\boldsymbol{a}_{2}}\) and the determinant of the matrix is 0, then they will produce degenerate parallelograms.

Question 7.
We learned that when multiplying -1 by a complex number z, for example, z=\(\left(\begin{array}{l}
3 \\
2
\end{array}\right)\), the resulting complex number z1=\(\left(\begin{array}{l}
-3 \\
-2
\end{array}\right)\) will be on the same line but on the opposite side of the origin. What matrix will produce the same effect? Verify your answer.
Answer:
The matrix that will rotate π radians is \(\left[\begin{array}{cc}
-1 & 0 \\
0 & -1
\end{array}\right]\).
\(\left(\begin{array}{cc}
-1 & 0 \\
0 & -1
\end{array}\right)\left(\begin{array}{l}
3 \\
2
\end{array}\right)\) = \(\left(\begin{array}{l}
-3 \\
-2
\end{array}\right)\)

Question 8.
A point z=\(\left(\begin{array}{l}
\sqrt{2} \\
\sqrt{2}
\end{array}\right)\) is transformed to \(\left(\begin{array}{c}
-2 \\
0
\end{array}\right)\). The final step of the transformation is adding the complex number \(\left(\begin{array}{c}
\mathbf{0} \\
-2
\end{array}\right)\). Describe a possible transformation that can get this result.
Answer:
Going backwards: \(\left(\begin{array}{c}
-2 \\
0
\end{array}\right)\)+(-1)\(\left(\begin{array}{c}
0 \\
-2
\end{array}\right)\)=\(\left(\begin{array}{c}
-2 \\
2
\end{array}\right)\). To move from \(\left(\begin{array}{l}
\sqrt{2} \\
\sqrt{2}
\end{array}\right)\) to \(\left(\begin{array}{c}
-2 \\
2
\end{array}\right)\) requires a \(\frac{\pi}{2}\) radians counterclockwise rotation: \(\left[\begin{array}{cc}
0 & -1 \\
1 & 0
\end{array}\right]\) and a dilation with a factor of \(\sqrt{2}\): \(\left[\begin{array}{cc}
\sqrt{2} & 0 \\
0 & \sqrt{2}
\end{array}\right]\).
Eureka Math Precalculus Module 2 Lesson 5 Problem Set Answer Key 60
Eureka Math Precalculus Module 2 Lesson 5 Problem Set Answer Key 61
Eureka Math Precalculus Module 2 Lesson 5 Problem Set Answer Key 62

Eureka Math Precalculus Module 2 Lesson 5 Exit Ticket Answer Key

Question 1.
Find the sum of the following, and plot the points and the resultant. Describe the geometric interpretation.
a. \(\left(\begin{array}{l}
3 \\
1
\end{array}\right)\)+\(\left(\begin{array}{l}
1 \\
3
\end{array}\right)\)
Eureka Math Precalculus Module 2 Lesson 5 Exit Ticket Answer Key 29
Answer:
\(\left(\begin{array}{l}
4 \\
4
\end{array}\right)\) The two points, the resultant, and the origin form a parallelogram.

b. \(\left(\begin{array}{l}
2 \\
0
\end{array}\right)\)+\(\left(\begin{array}{l}
1 \\
2
\end{array}\right)\)
Eureka Math Precalculus Module 2 Lesson 5 Exit Ticket Answer Key 30
Answer:
\(\left(\begin{array}{l}
3 \\
2
\end{array}\right)\) The two points, the resultant, and the origin form a parallelogram.

c. \(\left(\begin{array}{l}
-2 \\
4
\end{array}\right)\)+\(\left(\begin{array}{l}
3 \\
-2
\end{array}\right)\)
Eureka Math Precalculus Module 2 Lesson 5 Exit Ticket Answer Key 31
Answer:
\(\left(\begin{array}{l}
1 \\
2
\end{array}\right)\) The two points, the resultant, and the origin form a parallelogram.

d. –\(\left(\begin{array}{l}
3 \\
1
\end{array}\right)\)
Eureka Math Precalculus Module 2 Lesson 5 Exit Ticket Answer Key 32
Answer:
\(\left(\begin{array}{l}
-3 \\
-1
\end{array}\right)\) The resultant is a vector of the same magnitude in the opposite direction.

Question 2.
Find the sum of the following.
a. \(\left(\begin{array}{l}
3 \\
1 \\
3
\end{array}\right)\)+\(\left(\begin{array}{l}
1 \\
3 \\
1
\end{array}\right)\)
Answer:
\(\left(\begin{array}{l}
4 \\
4 \\
4
\end{array}\right)\)

b. \(\left(\begin{array}{l}
2 \\
0 \\
1
\end{array}\right)\)+\(\left(\begin{array}{l}
0 \\
2 \\
1
\end{array}\right)\)
Answer:
\(\left(\begin{array}{l}
2 \\
2 \\
2
\end{array}\right)\)

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