Linear Algebra Examples

$(5 i) (2 + 6 i)$

Step 1

Apply the distributive property.

$5 i \cdot 2 + 5 i (6 i)$

Step 2

Multiply $2$ by $5$ .

$10 i + 5 i (6 i)$

Step 3

Multiply $5 i (6 i)$ .

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Step 3.1

Multiply $6$ by $5$ .

$10 i + 30 i i$

Step 3.2

Raise $i$ to the power of $1$ .

$10 i + 30 (i^{1} i)$

Step 3.3

Raise $i$ to the power of $1$ .

$10 i + 30 (i^{1} i^{1})$

Step 3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$10 i + 30 i^{1 + 1}$

Step 3.5

Add $1$ and $1$ .

$10 i + 30 i^{2}$

Step 4

Simplify each term.

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Step 4.1

Rewrite $i^{2}$ as $- 1$ .

$10 i + 30 \cdot - 1$

Step 4.2

Multiply $30$ by $- 1$ .

$10 i - 30$

Step 5

Reorder $10 i$ and $- 30$ .

$- 30 + 10 i$

Step 6

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 7

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 8

Substitute the actual values of $a = - 30$ and $b = 10$ .

$| z | = \sqrt{10^{2} + {(- 30)}^{2}}$

Step 9

Find $| z |$ .

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Step 9.1

Raise $10$ to the power of $2$ .

$| z | = \sqrt{100 + {(- 30)}^{2}}$

Step 9.2

Raise $- 30$ to the power of $2$ .

$| z | = \sqrt{100 + 900}$

Step 9.3

Add $100$ and $900$ .

$| z | = \sqrt{1000}$

Step 9.4

Rewrite $1000$ as $10^{2} \cdot 10$ .

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Step 9.4.1

Factor $100$ out of $1000$ .

$| z | = \sqrt{100 (10)}$

Step 9.4.2

Rewrite $100$ as $10^{2}$ .

$| z | = \sqrt{10^{2} \cdot 10}$

Step 9.5

Pull terms out from under the radical.

$| z | = 10 \sqrt{10}$

Step 10

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

$θ = \arctan (\frac{10}{- 30})$

Step 11

Since inverse tangent of $\frac{10}{- 30}$ produces an angle in the second quadrant, the value of the angle is $2.81984209$ .

$θ = 2.81984209$

Step 12

Substitute the values of $θ = 2.81984209$ and $| z | = 10 \sqrt{10}$ .

$10 \sqrt{10} (\cos (2.81984209) + i \sin (2.81984209))$