Linear Algebra Examples

(1, 3)

$(1, 3)$

Step 1

To find a unit vector in the same direction as a vector

v⃗

$v⃗$ , divide by the norm of

v⃗

$v⃗$ .

\frac{v⃗}{| 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} + 3^{2}}

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

Step 3

Simplify.

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Step 3.1

One to any power is one.

\sqrt{1 + 3^{2}}

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

Step 3.2

Raise

3

$3$ to the power of

2

$2$ .

\sqrt{1 + 9}

$\sqrt{1 + 9}$

Step 3.3

Add

1

$1$ and

9

$9$ .

\sqrt{10}

$\sqrt{10}$

\sqrt{10}

$\sqrt{10}$

Step 4

Divide the vector by its norm.

\frac{(1, 3)}{\sqrt{10}}

$\frac{(1, 3)}{\sqrt{10}}$

Step 5

Divide each element in the vector by

\sqrt{10}

$\sqrt{10}$ .

(\frac{1}{\sqrt{10}}, \frac{3}{\sqrt{10}})

$(\frac{1}{\sqrt{10}}, \frac{3}{\sqrt{10}})$

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