Linear Algebra Examples

Find the Angle Between the Vectors Using Dot Product
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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
Subtract from .
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
One to any power is one.
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
Raise to the power of .
Step 6.1.2
Raise to the power of .
Step 6.1.3
Use the power rule to combine exponents.
Step 6.1.4
Add and .
Step 6.2
Rewrite as .
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Step 6.2.1
Use to rewrite as .
Step 6.2.2
Apply the power rule and multiply exponents, .
Step 6.2.3
Combine and .
Step 6.2.4
Cancel the common factor of .
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Step 6.2.4.1
Cancel the common factor.
Step 6.2.4.2
Rewrite the expression.
Step 6.2.5
Evaluate the exponent.
Step 6.3
Move the negative in front of the fraction.
Step 6.4
Evaluate .
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