# Linear Algebra Examples

,

Step 1

Use the dot product formula to find the angle between two vectors.

Step 2

Step 2.1

The dot product of two vectors is the sum of the products of the their components.

Step 2.2

Simplify.

Step 2.2.1

Simplify each term.

Step 2.2.1.1

Multiply by .

Step 2.2.1.2

Multiply by .

Step 2.2.2

Add and .

Step 3

Step 3.1

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

Step 3.2

Simplify.

Step 3.2.1

One to any power is one.

Step 3.2.2

Raising to any positive power yields .

Step 3.2.3

Add and .

Step 3.2.4

Any root of is .

Step 4

Step 4.1

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

Step 4.2

Simplify.

Step 4.2.1

Raising to any positive power yields .

Step 4.2.2

One to any power is one.

Step 4.2.3

Add and .

Step 4.2.4

Any root of is .

Step 5

Substitute the values into the formula.

Step 6

Step 6.1

Multiply by .

Step 6.2

Divide by .

Step 6.3

The exact value of is .