Linear Algebra Examples

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

Step 1

The distance between two vectors $u⃗$ and $v⃗$ in $ℝ^{n}$ 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 $u⃗ = [\begin{array}{c} 1 & 0 & 2 \end{array}]$ and $v⃗ = [\begin{array}{c} 1 & 1 & - 1 \end{array}]$ .

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Step 2.1

Create a vector of the difference.

$[\begin{array}{c} 1 - 1 \\ 0 - 1 \\ 2 + 1 \end{array}]$

Step 2.2

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

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

Step 2.3

Simplify.

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Step 2.3.1

Subtract $1$ from $1$ .

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

Step 2.3.2

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

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

Step 2.3.3

Subtract $1$ from $0$ .

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

Step 2.3.4

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

$\sqrt{0 + 1 + {(2 + 1)}^{2}}$

Step 2.3.5

Add $2$ and $1$ .

$\sqrt{0 + 1 + 3^{2}}$

Step 2.3.6

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

$\sqrt{0 + 1 + 9}$

Step 2.3.7

Add $0$ and $1$ .

$\sqrt{1 + 9}$

Step 2.3.8

Add $1$ and $9$ .

$\sqrt{10}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt{10}$

Decimal Form:

$3.16227766 \dots$

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