Precalculus Examples

Find the Angle Between the Vectors
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The equation for finding the angle between two vectors states that the dot product of the two vectors equals the product of the magnitudes of the vectors and the cosine of the angle between them.
Solve the equation for .
Find the dot product of the vectors.
To find the dot product, find the sum of the products of corresponding components of the vectors.
Substitute the components of the vectors into the expression.
Simplify.
Remove parentheses.
Simplify each term.
Multiply by .
Multiply by .
Find the magnitude of .
To find the magnitude of the vector, find the square root of the sum of the components of the vector squared.
Substitute the components of the vector into the expression.
Simplify.
Remove parentheses around .
One to any power is one.
Remove parentheses around .
Raise to the power of .
Find the magnitude of .
To find the magnitude of the vector, find the square root of the sum of the components of the vector squared.
Substitute the components of the vector into the expression.
Simplify.
Remove parentheses around .
Raise to the power of .
Remove parentheses around .
Raise to the power of .
Substitute the values into the equation for the angle between the vectors.
Simplify.
Remove parentheses.
Multiply by .
Multiply by .
Simplify the denominator.
Multiply by .
Combine using the product rule for radicals.
Simplify the denominator.
Multiply by .
Rewrite as .
Pull terms out from under the radical.
Reduce the expression by cancelling the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Multiply by .
Simplify.
Combine.
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.