Linear Algebra Examples

6 - (8 + 3 i)

$6 - (8 + 3 i)$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

6 - 1 \cdot 8 - (3 i)

$6 - 1 \cdot 8 - (3 i)$

Step 1.2

Multiply

- 1

$- 1$ by

8

$8$ .

6 - 8 - (3 i)

$6 - 8 - (3 i)$

Step 1.3

Multiply

3

$3$ by

- 1

$- 1$ .

6 - 8 - 3 i

$6 - 8 - 3 i$

6 - 8 - 3 i

$6 - 8 - 3 i$

Step 2

Subtract

8

$8$ from

6

$6$ .

- 2 - 3 i

$- 2 - 3 i$

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 = - 2

$a = - 2$ and

b = - 3

$b = - 3$ .

| z | = \sqrt{{(- 3)}^{2} + {(- 2)}^{2}}

$| z | = \sqrt{{(- 3)}^{2} + {(- 2)}^{2}}$

Step 6

Find

| z |

$| z |$ .

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Step 6.1

Raise

- 3

$- 3$ to the power of

2

$2$ .

| z | = \sqrt{9 + {(- 2)}^{2}}

$| z | = \sqrt{9 + {(- 2)}^{2}}$

Step 6.2

Raise

- 2

$- 2$ to the power of

2

$2$ .

| z | = \sqrt{9 + 4}

$| z | = \sqrt{9 + 4}$

Step 6.3

Add

9

$9$ and

4

$4$ .

| z | = \sqrt{13}

$| z | = \sqrt{13}$

| z | = \sqrt{13}

$| z | = \sqrt{13}$

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{- 3}{- 2})

$θ = \arctan (\frac{- 3}{- 2})$

Step 8

Since inverse tangent of

\frac{- 3}{- 2}

$\frac{- 3}{- 2}$ produces an angle in the third quadrant, the value of the angle is

4.12438637

$4.12438637$ .

θ = 4.12438637

$θ = 4.12438637$

Step 9

Substitute the values of

θ = 4.12438637

$θ = 4.12438637$ and

| z | = \sqrt{13}

$| z | = \sqrt{13}$ .

\sqrt{13} (cos (4.12438637) + i sin (4.12438637))

$\sqrt{13} (\cos (4.12438637) + i \sin (4.12438637))$