Linear Algebra Examples

2 + 7 i + (- 6 + i)

$2 + 7 i + (- 6 + i)$

Step 1

Subtract

6

$6$ from

2

$2$ .

- 4 + 7 i + i

$- 4 + 7 i + i$

Step 2

Add

7 i

$7 i$ and

i

$i$ .

- 4 + 8 i

$- 4 + 8 i$

Step 3

This is the trigonometric form of a complex number where

| z |

$| z |$ is the modulus and

θ

$θ$ is the angle created on the complex plane.

z = a + b i = | z | (cos (θ) + i sin (θ))

$z = a + b i = | z | (\cos (θ) + i \sin (θ))$

Step 4

The modulus of a complex number is the distance from the origin on the complex plane.

| z | = \sqrt{a^{2} + b^{2}}

$| z | = \sqrt{a^{2} + b^{2}}$ where

z = a + b i

$z = a + b i$

Step 5

Substitute the actual values of

a = - 4

$a = - 4$ and

b = 8

$b = 8$ .

| z | = \sqrt{8^{2} + {(- 4)}^{2}}

$| z | = \sqrt{8^{2} + {(- 4)}^{2}}$

Step 6

Find

| z |

$| z |$ .

Tap for more steps...

Step 6.1

Raise

8

$8$ to the power of

2

$2$ .

| z | = \sqrt{64 + {(- 4)}^{2}}

$| z | = \sqrt{64 + {(- 4)}^{2}}$

Step 6.2

Raise

- 4

$- 4$ to the power of

2

$2$ .

| z | = \sqrt{64 + 16}

$| z | = \sqrt{64 + 16}$

Step 6.3

Add

64

$64$ and

16

$16$ .

| z | = \sqrt{80}

$| z | = \sqrt{80}$

Step 6.4

Rewrite

80

$80$ as

4^{2} \cdot 5

$4^{2} \cdot 5$ .

Tap for more steps...

Step 6.4.1

Factor

16

$16$ out of

80

$80$ .

| z | = \sqrt{16 (5)}

$| z | = \sqrt{16 (5)}$

Step 6.4.2

Rewrite

16

$16$ as

4^{2}

$4^{2}$ .

| z | = \sqrt{4^{2} \cdot 5}

$| z | = \sqrt{4^{2} \cdot 5}$

| z | = \sqrt{4^{2} \cdot 5}

$| z | = \sqrt{4^{2} \cdot 5}$

Step 6.5

Pull terms out from under the radical.

| z | = 4 \sqrt{5}

$| z | = 4 \sqrt{5}$

| z | = 4 \sqrt{5}

$| z | = 4 \sqrt{5}$

Step 7

The angle of the point on the complex plane is the inverse tangent of the complex portion over the real portion.

θ = arctan (\frac{8}{- 4})

$θ = \arctan (\frac{8}{- 4})$

Step 8

Since inverse tangent of

\frac{8}{- 4}

$\frac{8}{- 4}$ produces an angle in the second quadrant, the value of the angle is

2.03444393

$2.03444393$ .

θ = 2.03444393

$θ = 2.03444393$

Step 9

Substitute the values of

θ = 2.03444393

$θ = 2.03444393$ and

| z | = 4 \sqrt{5}

$| z | = 4 \sqrt{5}$ .

4 \sqrt{5} (cos (2.03444393) + i sin (2.03444393))

$4 \sqrt{5} (\cos (2.03444393) + i \sin (2.03444393))$