# Precalculus 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

Raise to the power of .

Step 3.2.2

Raise to the power of .

Step 3.2.3

Add and .

Step 3.2.4

Rewrite as .

Step 3.2.4.1

Factor out of .

Step 3.2.4.2

Rewrite as .

Step 3.2.5

Pull terms out from under the radical.

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

Raise to the power of .

Step 4.2.2

Raise to the power of .

Step 4.2.3

Add and .

Step 5

Substitute the values into the formula.

Step 6

Step 6.1

Cancel the common factor of and .

Step 6.1.1

Factor out of .

Step 6.1.2

Cancel the common factors.

Step 6.1.2.1

Factor out of .

Step 6.1.2.2

Cancel the common factor.

Step 6.1.2.3

Rewrite the expression.

Step 6.2

Simplify the denominator.

Step 6.2.1

Combine using the product rule for radicals.

Step 6.2.2

Multiply by .

Step 6.3

Multiply by .

Step 6.4

Combine and simplify the denominator.

Step 6.4.1

Multiply by .

Step 6.4.2

Raise to the power of .

Step 6.4.3

Raise to the power of .

Step 6.4.4

Use the power rule to combine exponents.

Step 6.4.5

Add and .

Step 6.4.6

Rewrite as .

Step 6.4.6.1

Use to rewrite as .

Step 6.4.6.2

Apply the power rule and multiply exponents, .

Step 6.4.6.3

Combine and .

Step 6.4.6.4

Cancel the common factor of .

Step 6.4.6.4.1

Cancel the common factor.

Step 6.4.6.4.2

Rewrite the expression.

Step 6.4.6.5

Evaluate the exponent.

Step 6.5

Evaluate .