Linear Algebra Examples

$(2 + 8 i) - (6 + 9 i)$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$2 + 8 i - 1 \cdot 6 - (9 i)$

Step 1.2

Multiply $- 1$ by $6$ .

$2 + 8 i - 6 - (9 i)$

Step 1.3

Multiply $9$ by $- 1$ .

$2 + 8 i - 6 - 9 i$

Step 2

Simplify by adding terms.

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Step 2.1

Subtract $6$ from $2$ .

$- 4 + 8 i - 9 i$

Step 2.2

Subtract $9 i$ from $8 i$ .

$- 4 - i$

Step 3

This is the trigonometric form of a complex number where $| z |$ is the modulus and $θ$ is the angle created on the complex plane.

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

Step 4

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

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

Step 5

Substitute the actual values of $a = - 4$ and $b = - 1$ .

$| z | = \sqrt{{(- 1)}^{2} + {(- 4)}^{2}}$

Step 6

Find $| z |$ .

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Step 6.1

Raise $- 1$ to the power of $2$ .

$| z | = \sqrt{1 + {(- 4)}^{2}}$

Step 6.2

Raise $- 4$ to the power of $2$ .

$| z | = \sqrt{1 + 16}$

Step 6.3

Add $1$ and $16$ .

$| z | = \sqrt{17}$

Step 7

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

$θ = \arctan (\frac{- 1}{- 4})$

Step 8

Since inverse tangent of $\frac{- 1}{- 4}$ produces an angle in the third quadrant, the value of the angle is $3.38657131$ .

$θ = 3.38657131$

Step 9

Substitute the values of $θ = 3.38657131$ and $| z | = \sqrt{17}$ .

$\sqrt{17} (\cos (3.38657131) + i \sin (3.38657131))$