Linear Algebra Examples

$[\begin{array}{c} 2 + 0 i \\ 3 - 4 i \\ 4 + 3 i \end{array}]$

Step 1

The norm is the square root of the sum of squares of each element in the vector.

$\sqrt{{| 2 + 0 i |}^{2} + {| 3 - 4 i |}^{2} + {| 4 + 3 i |}^{2}}$

Step 2

Simplify.

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Step 2.1

Multiply $0$ by $i$ .

$\sqrt{{| 2 + 0 |}^{2} + {| 3 - 4 i |}^{2} + {| 4 + 3 i |}^{2}}$

Step 2.2

Add $2$ and $0$ .

$\sqrt{{| 2 |}^{2} + {| 3 - 4 i |}^{2} + {| 4 + 3 i |}^{2}}$

Step 2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\sqrt{2^{2} + {| 3 - 4 i |}^{2} + {| 4 + 3 i |}^{2}}$

Step 2.4

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

$\sqrt{4 + {| 3 - 4 i |}^{2} + {| 4 + 3 i |}^{2}}$

Step 2.5

Use the formula $| a + b i | = \sqrt{a^{2} + b^{2}}$ to find the magnitude.

$\sqrt{4 + {\sqrt{3^{2} + {(- 4)}^{2}}}^{2} + {| 4 + 3 i |}^{2}}$

Step 2.6

Raise $3$ to the power of $2$ .

$\sqrt{4 + {\sqrt{9 + {(- 4)}^{2}}}^{2} + {| 4 + 3 i |}^{2}}$

Step 2.7

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

$\sqrt{4 + {\sqrt{9 + 16}}^{2} + {| 4 + 3 i |}^{2}}$

Step 2.8

Add $9$ and $16$ .

$\sqrt{4 + {\sqrt{25}}^{2} + {| 4 + 3 i |}^{2}}$

Step 2.9

Rewrite $25$ as $5^{2}$ .

$\sqrt{4 + {\sqrt{5^{2}}}^{2} + {| 4 + 3 i |}^{2}}$

Step 2.10

Pull terms out from under the radical, assuming positive real numbers.

$\sqrt{4 + 5^{2} + {| 4 + 3 i |}^{2}}$

Step 2.11

Raise $5$ to the power of $2$ .

$\sqrt{4 + 25 + {| 4 + 3 i |}^{2}}$

Step 2.12

Use the formula $| a + b i | = \sqrt{a^{2} + b^{2}}$ to find the magnitude.

$\sqrt{4 + 25 + {\sqrt{4^{2} + 3^{2}}}^{2}}$

Step 2.13

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

$\sqrt{4 + 25 + {\sqrt{16 + 3^{2}}}^{2}}$

Step 2.14

Raise $3$ to the power of $2$ .

$\sqrt{4 + 25 + {\sqrt{16 + 9}}^{2}}$

Step 2.15

Add $16$ and $9$ .

$\sqrt{4 + 25 + {\sqrt{25}}^{2}}$

Step 2.16

Rewrite $25$ as $5^{2}$ .

$\sqrt{4 + 25 + {\sqrt{5^{2}}}^{2}}$

Step 2.17

Pull terms out from under the radical, assuming positive real numbers.

$\sqrt{4 + 25 + 5^{2}}$

Step 2.18

Raise $5$ to the power of $2$ .

$\sqrt{4 + 25 + 25}$

Step 2.19

Add $4$ and $25$ .

$\sqrt{29 + 25}$

Step 2.20

Add $29$ and $25$ .

$\sqrt{54}$

Step 2.21

Rewrite $54$ as $3^{2} \cdot 6$ .

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Step 2.21.1

Factor $9$ out of $54$ .

$\sqrt{9 (6)}$

Step 2.21.2

Rewrite $9$ as $3^{2}$ .

$\sqrt{3^{2} \cdot 6}$

Step 2.22

Pull terms out from under the radical.

$3 \sqrt{6}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$3 \sqrt{6}$

Decimal Form:

$7.34846922 \dots$