Linear Algebra Examples

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

Step 1

To find a unit vector in the same direction as a vector $v⃗$ , divide by the norm of $v⃗$ .

$\frac{v⃗}{| v⃗ |}$

Step 2

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

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

Step 3

Simplify.

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Step 3.1

One to any power is one.

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

Step 3.2

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

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

Step 3.3

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

$\sqrt{1 + 0 + 4}$

Step 3.4

Add $1$ and $0$ .

$\sqrt{1 + 4}$

Step 3.5

Add $1$ and $4$ .

$\sqrt{5}$

Step 4

Divide the vector by its norm.

$\frac{[\begin{array}{c} 1 \\ 0 \\ - 2 \end{array}]}{\sqrt{5}}$

Step 5

Divide each element in the vector by $\sqrt{5}$ .

$[\begin{array}{c} \frac{1}{\sqrt{5}} \\ \frac{0}{\sqrt{5}} \\ \frac{- 2}{\sqrt{5}} \end{array}]$

Step 6

Simplify.

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Step 6.1

Divide $0$ by $\sqrt{5}$ .

$[\begin{array}{c} \frac{1}{\sqrt{5}} \\ 0 \\ \frac{- 2}{\sqrt{5}} \end{array}]$

Step 6.2

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{1}{\sqrt{5}} \\ 0 \\ - \frac{2}{\sqrt{5}} \end{array}]$

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