# Linear Algebra Examples

,

Step 1

The distance between two vectors and in is defined to be which is the Euclidean norm of the difference .

Step 2

Step 2.1

Create a vector of the difference.

Step 2.2

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

Step 2.3

Simplify.

Step 2.3.1

Subtract from .

Step 2.3.2

Rearrange terms.

Step 2.3.3

Use the formula to find the magnitude.

Step 2.3.4

Raise to the power of .

Step 2.3.5

Raise to the power of .

Step 2.3.6

Add and .

Step 2.3.7

Rewrite as .

Step 2.3.7.1

Use to rewrite as .

Step 2.3.7.2

Apply the power rule and multiply exponents, .

Step 2.3.7.3

Combine and .

Step 2.3.7.4

Cancel the common factor of .

Step 2.3.7.4.1

Cancel the common factor.

Step 2.3.7.4.2

Rewrite the expression.

Step 2.3.7.5

Evaluate the exponent.

Step 2.3.8

Subtract from .

Step 2.3.9

Raise to the power of .

Step 2.3.10

Simplify each term.

Step 2.3.10.1

Apply the distributive property.

Step 2.3.10.2

Multiply by .

Step 2.3.10.3

Multiply by .

Step 2.3.10.4

Multiply by .

Step 2.3.11

Subtract from .

Step 2.3.12

Add and .

Step 2.3.13

Use the formula to find the magnitude.

Step 2.3.14

Raising to any positive power yields .

Step 2.3.15

One to any power is one.

Step 2.3.16

Add and .

Step 2.3.17

Any root of is .

Step 2.3.18

One to any power is one.

Step 2.3.19

Add and .

Step 2.3.20

Add and .

Step 3

The result can be shown in multiple forms.

Exact Form:

Decimal Form: