Linear Algebra Examples

$(- 7 + 4 i) - (- 9 - 3 i)$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$- 7 + 4 i - - 9 - (- 3 i)$

Step 1.2

Multiply $- 1$ by $- 9$ .

$- 7 + 4 i + 9 - (- 3 i)$

Step 1.3

Multiply $- 3$ by $- 1$ .

$- 7 + 4 i + 9 + 3 i$

Step 2

Simplify by adding terms.

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Step 2.1

Add $- 7$ and $9$ .

$2 + 4 i + 3 i$

Step 2.2

Add $4 i$ and $3 i$ .

$2 + 7 i$

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 $a = 2$ and $b = 7$ .

$| z | = \sqrt{7^{2} + 2^{2}}$

Step 6

Find $| z |$ .

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Step 6.1

Raise $7$ to the power of $2$ .

$| z | = \sqrt{49 + 2^{2}}$

Step 6.2

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

$| z | = \sqrt{49 + 4}$

Step 6.3

Add $49$ and $4$ .

$| z | = \sqrt{53}$

Step 7

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

$θ = \arctan (\frac{7}{2})$

Step 8

Since inverse tangent of $\frac{7}{2}$ produces an angle in the first quadrant, the value of the angle is $1.29249666$ .

$θ = 1.29249666$

Step 9

Substitute the values of $θ = 1.29249666$ and $| z | = \sqrt{53}$ .

$\sqrt{53} (\cos (1.29249666) + i \sin (1.29249666))$