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.
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Step 2.1
The dot product of two vectors is the sum of the products of the their components.
Step 2.2
Simplify.
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Step 2.2.1
Simplify each term.
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Step 2.2.1.1
Multiply by .
Step 2.2.1.2
Multiply by .
Step 2.2.2
Add and .
Step 3
Find the magnitude of .
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Step 3.1
The norm is the square root of the sum of squares of each element in the vector.
Step 3.2
Simplify.
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Step 3.2.1
One to any power is one.
Step 3.2.2
Raise to the power of .
Step 3.2.3
Add and .
Step 4
Find the magnitude of .
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Step 4.1
The norm is the square root of the sum of squares of each element in the vector.
Step 4.2
Simplify.
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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
Simplify.
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Step 6.1
Simplify the denominator.
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Step 6.1.1
Combine using the product rule for radicals.
Step 6.1.2
Multiply by .
Step 6.2
Simplify the denominator.
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Step 6.2.1
Rewrite as .
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Step 6.2.1.1
Factor out of .
Step 6.2.1.2
Rewrite as .
Step 6.2.2
Pull terms out from under the radical.
Step 6.3
Cancel the common factor of and .
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Step 6.3.1
Factor out of .
Step 6.3.2
Cancel the common factors.
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Step 6.3.2.1
Factor out of .
Step 6.3.2.2
Cancel the common factor.
Step 6.3.2.3
Rewrite the expression.
Step 6.4
Multiply by .
Step 6.5
Combine and simplify the denominator.
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Step 6.5.1
Multiply by .
Step 6.5.2
Raise to the power of .
Step 6.5.3
Raise to the power of .
Step 6.5.4
Use the power rule to combine exponents.
Step 6.5.5
Add and .
Step 6.5.6
Rewrite as .
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Step 6.5.6.1
Use to rewrite as .
Step 6.5.6.2
Apply the power rule and multiply exponents, .
Step 6.5.6.3
Combine and .
Step 6.5.6.4
Cancel the common factor of .
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Step 6.5.6.4.1
Cancel the common factor.
Step 6.5.6.4.2
Rewrite the expression.
Step 6.5.6.5
Evaluate the exponent.
Step 6.6
Evaluate .
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