Linear Algebra Examples

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

Step 1

Subtract $6$ from $2$ .

$- 4 + 7 i + i$

Step 2

Add $7 i$ and $i$ .

$- 4 + 8 i$

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 $b = 8$ .

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

Step 6

Find $| z |$ .

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Step 6.1

Raise $8$ to the power of $2$ .

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

Step 6.2

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

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

Step 6.3

Add $64$ and $16$ .

$| z | = \sqrt{80}$

Step 6.4

Rewrite $80$ as $4^{2} \cdot 5$ .

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Step 6.4.1

Factor $16$ out of $80$ .

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

Step 6.4.2

Rewrite $16$ as $4^{2}$ .

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

Step 6.5

Pull terms out from under the radical.

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

Step 7

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

$θ = \arctan (\frac{8}{- 4})$

Step 8

Since inverse tangent of $\frac{8}{- 4}$ produces an angle in the second quadrant, the value of the angle is $2.03444393$ .

$θ = 2.03444393$

Step 9

Substitute the values of $θ = 2.03444393$ and $| z | = 4 \sqrt{5}$ .

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