# Precalculus Examples

Find the Angle Between the Vectors
,
Step 1
Use the dot product formula to find the angle between two vectors.
Step 2
Find the dot product.
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
Step 3
Find the magnitude of .
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
Step 4
Find the magnitude of .
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
Step 4.2.4
Rewrite as .
Step 4.2.5
Pull terms out from under the radical, assuming positive real numbers.
Step 5
Substitute the values into the formula.
Step 6
Simplify.
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
Move to the left of .
Step 6.3
Multiply by .
Step 6.4
Combine and simplify the denominator.
Step 6.4.1
Multiply by .
Step 6.4.2
Move .
Step 6.4.3
Raise to the power of .
Step 6.4.4
Raise to the power of .
Step 6.4.5
Use the power rule to combine exponents.
Step 6.4.6
Step 6.4.7
Rewrite as .
Step 6.4.7.1
Use to rewrite as .
Step 6.4.7.2
Apply the power rule and multiply exponents, .
Step 6.4.7.3
Combine and .
Step 6.4.7.4
Cancel the common factor of .
Step 6.4.7.4.1
Cancel the common factor.
Step 6.4.7.4.2
Rewrite the expression.
Step 6.4.7.5
Evaluate the exponent.
Step 6.5
Multiply by .
Step 6.6
Evaluate .