# Eureka Math Precalculus Module 2 Lesson 5 Answer Key

## Engage NY Eureka Math Precalculus Module 2 Lesson 5 Answer Key

### Eureka Math Precalculus Module 2 Lesson 5 Exercise Answer Key

Opening Exercise

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

b. 5∙(2+3i)
10+15i

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

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

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.

$$\left(\begin{array}{l} 7 \\ 4 \end{array}\right)$$

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.

$$\left(\begin{array}{c} -3 \\ 5 \end{array}\right)$$

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.

$$\left(\begin{array}{c} 2 \\ -1 \end{array}\right)$$

Exercise 4.
Let x=$$\left(\begin{array}{c} 3 \\ 2 \end{array}\right)$$. Compute z=2x, and plot x and z in the plane.

$$\left(\begin{array}{l} 6 \\ 4 \end{array}\right)$$

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.

$$\left(\begin{array}{c} 2 \\ -1 \end{array}\right)$$

Exercise 6.
Let x=$$\left(\begin{array}{c} 1 \\ -1 \end{array}\right)$$. Compute z=-3x, and plot x and z in the plane.

$$\left(\begin{array}{c} -3 \\ 3 \end{array}\right)$$

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.

$$\left(\begin{array}{l} 4 \\ 4 \\ 2 \end{array}\right)$$

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.

$$\left(\begin{array}{l} 3 \\ 3 \\ 0 \end{array}\right)$$

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.

$$\left(\begin{array}{l} 4 \\ 4 \\ 4 \end{array}\right)$$

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.

$$\left(\begin{array}{l} -1 \\ -2 \\ -2 \end{array}\right)$$
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=$$\left(\begin{array}{l} 3 \\ 2 \end{array}\right)$$

$$\left(\begin{array}{l} 5 \\ 5 \end{array}\right)$$ The two points, the resultant, and the origin form a parallelogram.

b. x=$$\left(\begin{array}{l} 2 \\ 4 \end{array}\right)$$, y=$$\left(\begin{array}{l} -4 \\ 2 \end{array}\right)$$

$$\left(\begin{array}{l} -2 \\ 6 \end{array}\right)$$ The two points, the resultant, and the origin form a parallelogram.

c. x=$$\left(\begin{array}{l} 2 \\ 1 \end{array}\right)$$, y=$$\left(\begin{array}{l} -4 \\ -2 \end{array}\right)$$

$$\left(\begin{array}{l} -2 \\ -1 \end{array}\right)$$ 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=$$\left(\begin{array}{l} 2 \\ 4 \end{array}\right)$$

$$\left(\begin{array}{l} 3 \\ 6 \end{array}\right)$$ 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= ?

$$\left(\begin{array}{l} 2 \\ 4 \end{array}\right)$$ 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= ?

$$\left(\begin{array}{l} 2 \\ -1 \end{array}\right)$$ 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= ?

$$\left(\begin{array}{c} 0 \\ 0 \end{array}\right)$$ 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=$$\left(\begin{array}{l} 2 \\ 1 \\ 2 \end{array}\right)$$

$$\left(\begin{array}{l} 3 \\ 2 \\ 4 \end{array}\right)$$ 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=$$\left(\begin{array}{l} 2 \\ 2 \\ 2 \end{array}\right)$$

$$\left(\begin{array}{l} 3 \\ 3 \\ 3 \end{array}\right)$$ Three points all collapse onto the same line in space.

c. x=$$\left(\begin{array}{l} 2 \\ 0 \\ 0 \end{array}\right)$$, y=$$\left(\begin{array}{l} 0 \\ 2 \\ 0 \end{array}\right)$$

$$\left(\begin{array}{l} 2 \\ 2 \\ 0 \end{array}\right)$$ 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= ?

$$\left(\begin{array}{l} 4 \\ 2 \\ 2 \end{array}\right)$$ The point is dilated by a factor of 2.

b. x=$$\left(\begin{array}{l} 2 \\ 2 \\ 2 \end{array}\right)$$, k=-1, kx= ?

$$\left(\begin{array}{c} -2 \\ -2 \\ -2 \end{array}\right)$$ 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 $$\left(\begin{array}{l} 0 \\ 0 \end{array}\right)$$
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 $$\left(\begin{array}{l} 0 \\ 0 \\ 0 \end{array}\right)$$
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.
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
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
$$\left(\begin{array}{c} -\frac{1}{2} \\ \frac{3}{2} \end{array}\right)$$

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

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.

$$\left(\begin{array}{l} -2 \\ -1 \end{array}\right)$$

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.

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

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

e. What is the determinant of the matrix $$\left(\begin{array}{ll} 2 & -4 \\ 1 & -2 \end{array}\right)$$?
$$\left|\begin{array}{ll} 2 & -4 \\ 1 & -2 \end{array}\right|$$=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)?
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.
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.
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 Exit Ticket Answer Key

Question 1.
Find the sum of the following, and plot the points and the resultant. Describe the geometric interpretation.
a. $$\left(\begin{array}{l} 3 \\ 1 \end{array}\right)$$+$$\left(\begin{array}{l} 1 \\ 3 \end{array}\right)$$

$$\left(\begin{array}{l} 4 \\ 4 \end{array}\right)$$ The two points, the resultant, and the origin form a parallelogram.

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

$$\left(\begin{array}{l} 3 \\ 2 \end{array}\right)$$ The two points, the resultant, and the origin form a parallelogram.

c. $$\left(\begin{array}{l} -2 \\ 4 \end{array}\right)$$+$$\left(\begin{array}{l} 3 \\ -2 \end{array}\right)$$

$$\left(\begin{array}{l} 1 \\ 2 \end{array}\right)$$ The two points, the resultant, and the origin form a parallelogram.

d. –$$\left(\begin{array}{l} 3 \\ 1 \end{array}\right)$$

$$\left(\begin{array}{l} -3 \\ -1 \end{array}\right)$$ The resultant is a vector of the same magnitude in the opposite direction.
a. $$\left(\begin{array}{l} 3 \\ 1 \\ 3 \end{array}\right)$$+$$\left(\begin{array}{l} 1 \\ 3 \\ 1 \end{array}\right)$$
$$\left(\begin{array}{l} 4 \\ 4 \\ 4 \end{array}\right)$$
b. $$\left(\begin{array}{l} 2 \\ 0 \\ 1 \end{array}\right)$$+$$\left(\begin{array}{l} 0 \\ 2 \\ 1 \end{array}\right)$$
$$\left(\begin{array}{l} 2 \\ 2 \\ 2 \end{array}\right)$$