## Engage NY Eureka Math Precalculus Module 1 Lesson 6 Answer Key

### Eureka Math Precalculus Module 1 Lesson 6 Exercise Answer Key

Opening Exercise

Perform the indicated arithmetic operations for complex numbers z = -4 + 5i and w = -1-2i.

a. z + w

Answer:

z + w = -5 + 3i

b. z-w

Answer:

z-w = -3 + 7i

c.z + 2w

Answer:

z + 2w = -6 + i

d.z-z

Answer:

z-z = 0 + 0i

e. Explain how you add and subtract complex numbers.

Answer:

Add or subtract the real components and the imaginary components separately.

Exercises

Exercise 1.

The length of the vector that represents z_{1} = 6-8i is 10 because \(\sqrt{6^{2} + (-8)^{2}}\) = \(\sqrt{100}\) = 10.

a. Find at least seven other complex numbers that can be represented as vectors that have length 10.

Answer:

There are an infinite number of complex numbers that meet this criteria; the most obvious are 10, 6 + 8i, 8 + 6i, 10i, -6 + 8i, -8 + 6i, -10, -8-6i, -6-8i, -10i, 8-6i, and -8 + 6i. The associated vectors for these numbers are shown in the sample response for part (b).

b. Draw the vectors on the coordinate axes provided below.

Answer:

c. What do you observe about all of these vectors?

Answer:

Students should observe that the tips of the vectors lie on the circle of radius 10 centered at the origin.

Exercise 2.

In the Opening Exercise, we computed z + 2w. Calculate this sum using vectors.

Answer:

Exercise 3.

In the Opening Exercise, we also computed z-z. Calculate this sum using vectors.

Answer:

Exercise 4.

For the vectors u and v pictured below, draw the specified sum or difference on the coordinate axes provided.

a. u + v

b. v-u

c. 2u-v

d. -u-3v

Answer:

Exercise 5.

Find the sum of 4 + i and -3 + 2i geometrically.

Answer:

1 + 3i

Exercise 6.

Show that (7 + 2i)-(4-i) = 3 + 3i by representing the complex numbers as vectors.

Answer:

### Eureka Math Precalculus Module 1 Lesson 6 Problem Set Answer Key

Question 1.

Let z = 1 + i and w = 1-3i. Find the following. Express your answers in a + bi form.

a. z + w

Answer:

1 + i + 1-3i = 2-2i

b. z-w

Answer:

1 + i-(1-3i) = 1 + i-1 + 3i

= 0 + 4i

c. 4w

Answer:

4(1-3i) = 4-12i

d. 3z + w

Answer:

3(1 + i) + 1-3i = 3 + 3i + 1-3i

= 4 + 0i

e. -w-2z

Answer:

-(1-3i)-2(1 + i) = -1 + 3i-2-2i

= -3 + i

f. What is the length of the vector representing z?

Answer:

The length of the vector representing z is \(\sqrt{1^{2} + 1^{2}}\) = \(\sqrt{2}\).

g. What is the length of the vector representing w?

Answer:

The length of the vector representing w is \(\sqrt{1^{2} + (-3)^{2}}\) = \(\sqrt{10}\).

Question 2.

Let u = 3 + 2i, v = 1 + i, and w = -2-i. Find the following. Express your answer in a + bi form, and represent the result in the plane.

a. u-2v

Answer:

3 + 2i-2(1 + i) = 3 + 2i-2-2i

= 1 + 0i

b. u-2w

Answer:

3 + 2i-2(-2-i) = 3 + 2i + 4 + 2i

= 7 + 4i

c. u + v + w

Answer:

3 + 2i + 1 + i-2-i = 2 + 2i

d. u-v + w

Answer:

3 + 2i-(1 + i)-2-i = 3 + 2i-1-i-2-i

= 0 + 0i

e. What is the length of the vector representing u?

Answer:

The length of the vector representing u is \(\sqrt{3^{2} + 2^{2}}\) = \(\sqrt{13}\).

f. What is the length of the vector representing u-v + w?

Answer:

The length of the vector representing u-v + w = \(\sqrt{0^{2} + 0^{2}}\) = \(\sqrt{0}\) = 0.

Question 3.

Find the sum of -2-4i and 5 + 3i geometrically.

Answer:

3-i

Question 4.

Show that (-5-6i)-(-8-4i) = 3-2i by representing the complex numbers as vectors.

Answer:

Question 5.

Let z_{1} = a_{1} + b_{1} i, z_{2} = a_{2} + b_{2} i, and z_{3} = a_{3} + b_{3} i. Prove the following using algebra or by showing with vectors.

Answer:

z_{1} + z_{2} = z_{2} + z_{1}

z_{1} + z_{2} = (a_{1} + b_{1} i) + (a_{2} + b_{2} i)

= (a_{2} + b_{2} i) + (a_{1} + b_{1} i)

= z_{2} + z_{1}

b. z_{1} + (z_{2} + z_{3} ) = (z_{1} + z_{2} ) + z_{3}

Answer:

z_{1} + (z_{2} + z_{3} ) = (a_{1} + b_{1} i) + ((a_{2} + b_{2} i) + (a_{3} + b_{3} i))

= ((a_{1} + b_{1} i) + (a_{2} + b_{2} i)) + (a_{3} + b_{3} i)

= (z_{1} + z_{2} ) + z_{3}

Question 6.

Let z = -3-4i and w = -3 + 4i.

a. Draw vectors representing z and w on the same set of axes.

Answer:

b. What are the lengths of the vectors representing z and w?

Answer:

The length of the vector representing z is \(\sqrt{(-3)^{2} + (-4)^{2}}\) = \(\sqrt{25}\) = 5.

The length of the vector representing w is \(\sqrt{(-3)^{2} + 4^{2}}\) = \(\sqrt{25}\) = 5.

c. Find a new vector, u_{z}, such that u_{z} is equal to z divided by the length of the vector representing z.

Answer:

u_{z} = \(\frac{-3-4 i}{5}\) = \(\frac{-3}{5}\) – \(\frac{4}{5}\)i

d. Find u_{w}, such that u_{w} is equal to w divided by the length of the vector representing w.

Answer:

u_{w} = \(\frac{-3 + 4 i}{5}\) = \(\frac{-3}{5}\) + \(\frac{4}{5}\)i

e. Draw vectors representing u_{z} and u_{w} on the same set of axes as part (a).

Answer:

f. What are the lengths of the vectors representing u_{z} and u_{w}?

Answer:

The length of the vector representing u_{z} is \(\sqrt{\left(-\frac{3}{5}\right)^{2} + \left(-\frac{4}{5}\right)^{2}}\) = \(\sqrt{\frac{9}{25} + \frac{16}{25}}\) = \(\sqrt{\frac{25}{25}}\) = \(\sqrt{1}\) = 1.

The length of the vector representing u_{w} is \(\sqrt{\left(-\frac{3}{5}\right)^{2} + \left(\frac{4}{5}\right)^{2}}\) = \(\sqrt{\frac{9}{25} + \frac{16}{25}}\) = \(\sqrt{\frac{25}{25}}\) = \(\sqrt{1}\) = 1.

g. Compare the vectors representing u_{z} to z and u_{w} to w. What do you notice?

Answer:

The vectors representing u_{z} and u_{w} are in the same direction as z and w, respectively, but their lengths are only 1.

h. What is the value of u_{z} times u_{w}?

Answer:

(\(\frac{3}{5}\)–\(\frac{4}{5}\)i)(\(\frac{3}{5}\))^{2}-(\(\frac{4}{5}\)i)^{2} = (\(\frac{9}{25}\)) + (\(\frac{16}{25}\)) = 1

i. What does your answer to part (h) tell you about the relationship between u_{z} and u_{w}?

Answer:

Since their product is 1, we know that u_{z} and u_{w} are reciprocals of each other.

Question 7.

Let z = a + bi.

a. Let u_{z} be represented by the vector in the direction of z with length 1. How can you find u_{z}? What is the value of u_{z}?

Answer:

Find the length of z, and then divide z by its length.

u_{z} = \(\frac{a + b i}{\sqrt{a^{2} + b^{2}}}\)

b. Let u_{w} be the complex number that when multiplied by u_{z}, the product is 1. What is the value of u_{w}?

Answer:

From Problem 4, we expect u_{w} = \(\frac{a-b i}{\sqrt{a^{2} + b^{2}}}\). Multiplying, we get

\(\frac{a + b i}{\sqrt{a^{2} + b^{2}}}\) âˆ™ \(\frac{a-b i}{\sqrt{a^{2} + b^{2}}}\) = \(\frac{a^{2}-(b i)^{2}}{a^{2} + b^{2}}\)

= \(\frac{a^{2} + b^{2}}{a^{2} + b^{2}}\)

= 1

c. What number could we multiply z by to get a product of 1?

Answer:

Since we know that u_{z} is equal to z divided by the length of z and that u_{z} âˆ™ u_{w} = 1, we get

z âˆ™ \(\frac{1}{\sqrt{a^{2} + b^{2}}}\)âˆ™\(\frac{a-b i}{\sqrt{a^{2} + b^{2}}}\) = zâˆ™\(\frac{a-b i}{a^{2} + b^{2}}\) = 1

So, multiplying z by \(\frac{a-b i}{a^{2} + b^{2}}\) will result in a product of 1.

Question 8.

Let z = -3 + 5i.

a. Draw a picture representing z + w = 8 + 2i.

Answer:

b. What is the value of w?

Answer:

w = 11 – 3i

### Eureka Math Precalculus Module 1 Lesson 6 Exit Ticket Answer Key

Answer:

Let z = -1 + 2i and w = 2 + i. Find the following, and verify each geometrically by graphing z, w, and each result.

a. z + w

Answer:

1 + 3i

b. z-w

Answer:

-3 + i

c. 2z-w

Answer:

-4 + 3i

d. w-z

Answer:

3-i