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.
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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.
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Remove parentheses.
Simplify each term.
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Multiply by .
Multiply by .
Add and .
Find the magnitude of .
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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.
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Raise to the power of .
Raise to the power of .
Add and .
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Find the magnitude of .
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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.
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Raise to the power of .
Raise to the power of .
Add and .
Rewrite as .
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Factor out of .
Rewrite as .
Pull terms out from under the radical.
Substitute the values into the equation for the angle between the vectors.
Simplify.
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Cancel the common factor of and .
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Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Cancel the common factor of and .
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Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Multiply by .
Combine and simplify the denominator.
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Multiply and .
Move .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Add and .
Rewrite as .
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Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Evaluate the exponent.
Multiply by .
Evaluate .
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