Linear Algebra Examples

$[\begin{array}{c} 2 i - 3 & 0 & 2 \end{array}]$ , $[\begin{array}{c} 0 & 1 & 2 - i \end{array}]$

Step 1

The distance between two vectors $u⃗$ and $v⃗$ in $ℂ^{n}$ is defined to be $| | u⃗ - v⃗ | |$ which is the Euclidean norm of the difference $u⃗ - v⃗$ .

$d (u⃗, v⃗) = | | u⃗ - v⃗ | | = \sqrt{{| {u⃗}_{1} - {v⃗}_{1} |}^{2} + {| {u⃗}_{2} - {v⃗}_{2} |}^{2} + \dots + {| {u⃗}_{n} - {v⃗}_{n} |}^{2}}$

Step 2

Find the norm of the difference $u⃗ - v⃗$ where $u⃗ = [\begin{array}{c} 2 i - 3 & 0 & 2 \end{array}]$ and $v⃗ = [\begin{array}{c} 0 & 1 & 2 - i \end{array}]$ .

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Step 2.1

Create a vector of the difference.

$[\begin{array}{c} 2 i - 3 - 0 \\ 0 - 1 \\ 2 - (2 - i) \end{array}]$

Step 2.2

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

$\sqrt{{| 2 i - 3 - 0 |}^{2} + {(0 - 1)}^{2} + {| 2 - (2 - i) |}^{2}}$

Step 2.3

Simplify.

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Step 2.3.1

Subtract $0$ from $2 i - 3$ .

$\sqrt{{| 2 i - 3 |}^{2} + {(0 - 1)}^{2} + {| 2 - (2 - i) |}^{2}}$

Step 2.3.2

Rearrange terms.

$\sqrt{{| - 3 + 2 i |}^{2} + {(0 - 1)}^{2} + {| 2 - (2 - i) |}^{2}}$

Step 2.3.3

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

$\sqrt{{\sqrt{{(- 3)}^{2} + 2^{2}}}^{2} + {(0 - 1)}^{2} + {| 2 - (2 - i) |}^{2}}$

Step 2.3.4

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

$\sqrt{{\sqrt{9 + 2^{2}}}^{2} + {(0 - 1)}^{2} + {| 2 - (2 - i) |}^{2}}$

Step 2.3.5

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

$\sqrt{{\sqrt{9 + 4}}^{2} + {(0 - 1)}^{2} + {| 2 - (2 - i) |}^{2}}$

Step 2.3.6

Add $9$ and $4$ .

$\sqrt{{\sqrt{13}}^{2} + {(0 - 1)}^{2} + {| 2 - (2 - i) |}^{2}}$

Step 2.3.7

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 2.3.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\sqrt{{(13^{\frac{1}{2}})}^{2} + {(0 - 1)}^{2} + {| 2 - (2 - i) |}^{2}}$

Step 2.3.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{13^{\frac{1}{2} \cdot 2} + {(0 - 1)}^{2} + {| 2 - (2 - i) |}^{2}}$

Step 2.3.7.3

Combine $\frac{1}{2}$ and $2$ .

$\sqrt{13^{\frac{2}{2}} + {(0 - 1)}^{2} + {| 2 - (2 - i) |}^{2}}$

Step 2.3.7.4

Cancel the common factor of $2$ .

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Step 2.3.7.4.1

Cancel the common factor.

$\sqrt{13^{\frac{2}{2}} + {(0 - 1)}^{2} + {| 2 - (2 - i) |}^{2}}$

Step 2.3.7.4.2

Rewrite the expression.

$\sqrt{13^{1} + {(0 - 1)}^{2} + {| 2 - (2 - i) |}^{2}}$

Step 2.3.7.5

Evaluate the exponent.

$\sqrt{13 + {(0 - 1)}^{2} + {| 2 - (2 - i) |}^{2}}$

Step 2.3.8

Subtract $1$ from $0$ .

$\sqrt{13 + {(- 1)}^{2} + {| 2 - (2 - i) |}^{2}}$

Step 2.3.9

Raise $- 1$ to the power of $2$ .

$\sqrt{13 + 1 + {| 2 - (2 - i) |}^{2}}$

Step 2.3.10

Simplify each term.

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Step 2.3.10.1

Apply the distributive property.

$\sqrt{13 + 1 + {| 2 - 1 \cdot 2 - - i |}^{2}}$

Step 2.3.10.2

Multiply $- 1$ by $2$ .

$\sqrt{13 + 1 + {| 2 - 2 - - i |}^{2}}$

Step 2.3.10.3

Multiply $- 1$ by $- 1$ .

$\sqrt{13 + 1 + {| 2 - 2 + 1 i |}^{2}}$

Step 2.3.10.4

Multiply $i$ by $1$ .

$\sqrt{13 + 1 + {| 2 - 2 + i |}^{2}}$

Step 2.3.11

Subtract $2$ from $2$ .

$\sqrt{13 + 1 + {| 0 + i |}^{2}}$

Step 2.3.12

Add $0$ and $i$ .

$\sqrt{13 + 1 + {| i |}^{2}}$

Step 2.3.13

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

$\sqrt{13 + 1 + {\sqrt{0^{2} + 1^{2}}}^{2}}$

Step 2.3.14

Raising $0$ to any positive power yields $0$ .

$\sqrt{13 + 1 + {\sqrt{0 + 1^{2}}}^{2}}$

Step 2.3.15

One to any power is one.

$\sqrt{13 + 1 + {\sqrt{0 + 1}}^{2}}$

Step 2.3.16

Add $0$ and $1$ .

$\sqrt{13 + 1 + {\sqrt{1}}^{2}}$

Step 2.3.17

Any root of $1$ is $1$ .

$\sqrt{13 + 1 + 1^{2}}$

Step 2.3.18

One to any power is one.

$\sqrt{13 + 1 + 1}$

Step 2.3.19

Add $13$ and $1$ .

$\sqrt{14 + 1}$

Step 2.3.20

Add $14$ and $1$ .

$\sqrt{15}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt{15}$

Decimal Form:

$3.87298334 \dots$

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