[[two plus zero i],[three minus four i],[four plus three i]]

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

The norm of a vector is the square root of the sum of each element of the vector squared.

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

Clarify each term.

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Initially, First, multiply $0$ by $i$ .

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

Add $2$ and $0$ .

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

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

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

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

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

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

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

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

Add $9$ and $16$ .

Should be rewritten $25$ as $5^{2}$ .

Withdraw terms from under the radical, assuming positive real numbers.

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

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

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

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

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

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

Add $16$ and $9$ .

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

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

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

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

$\sqrt{4 + 25 + 25}$

Clarify by adding numbers.

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Add $4$ and $25$ .

$\sqrt{29 + 25}$

Add $29$ and $25$ .

$\sqrt{54}$

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

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Factor $9$ out of $54$ .

$\sqrt{9 (6)}$

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

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

Pull terms out from under the radical.

$3 \sqrt{6}$

The result can be displayed in a variety of ways.

Exact Form:

$3 \sqrt{6}$

Decimal Form:

$7.34846922 \dots$

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