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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