# Linear Algebra Examples

Find the Distance Between the Vectors
,
Step 1
The distance between two vectors and in is defined to be which is the Euclidean norm of the difference .
Step 2
Find the norm of the difference where and .
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
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
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
Step 2.3.17
Any root of is .
Step 2.3.18
One to any power is one.
Step 2.3.19