Find the Angle Between the Vectors (5,5) , (4,5)

Math
(5,5) , (4,5)
The equation for finding the angle between two vectors θ 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.
u⋅v=|u||v|cos(θ)
Solve the equation for θ.
θ=arc⋅cos(u⋅v|u||v|)
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.
u⋅v=u1v1+u2v2
Substitute the components of the vectors into the expression.
5⋅4+5⋅5
Simplify.
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Remove parentheses.
5⋅4+5⋅5
Simplify each term.
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Multiply 5 by 4.
20+5⋅5
Multiply 5 by 5.
20+25
20+25
Add 20 and 25.
45
45
45
Find the magnitude of u.
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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.
(u1)2+(u2)2
Substitute the components of the vector into the expression.
(5)2+(5)2
Simplify.
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Raise 5 to the power of 2.
25+(5)2
Raise 5 to the power of 2.
25+25
Add 25 and 25.
50
Rewrite 50 as 52⋅2.
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Factor 25 out of 50.
25(2)
Rewrite 25 as 52.
52⋅2
52⋅2
Pull terms out from under the radical.
52
52
52
Find the magnitude of v.
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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.
(u1)2+(u2)2
Substitute the components of the vector into the expression.
(4)2+(5)2
Simplify.
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Raise 4 to the power of 2.
16+(5)2
Raise 5 to the power of 2.
16+25
Add 16 and 25.
41
41
41
Substitute the values into the equation for the angle between the vectors.
θ=arccos(45(52)⋅(41))
Simplify.
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Cancel the common factor of 45 and 5.
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Factor 5 out of 45.
arccos(5⋅952⋅41)
Cancel the common factors.
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Factor 5 out of 52⋅41.
arccos(5⋅95(2⋅41))
Cancel the common factor.
arccos(5⋅95(2⋅41))
Rewrite the expression.
arccos(92⋅41)
arccos(92⋅41)
arccos(92⋅41)
Simplify the denominator.
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Combine using the product rule for radicals.
arccos(92⋅41)
Multiply 2 by 41.
arccos(982)
arccos(982)
Multiply 982 by 8282.
arccos(982⋅8282)
Combine and simplify the denominator.
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Multiply 982 and 8282.
arccos(9828282)
Raise 82 to the power of 1.
arccos(98282182)
Raise 82 to the power of 1.
arccos(982821821)
Use the power rule aman=am+n to combine exponents.
arccos(982821+1)
Add 1 and 1.
arccos(982822)
Rewrite 822 as 82.
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Use axn=axn to rewrite 82 as 8212.
arccos(982(8212)2)
Apply the power rule and multiply exponents, (am)n=amn.
arccos(9828212⋅2)
Combine 12 and 2.
arccos(9828222)
Cancel the common factor of 2.
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Cancel the common factor.
arccos(9828222)
Divide 1 by 1.
arccos(982821)
arccos(982821)
Evaluate the exponent.
arccos(98282)
arccos(98282)
arccos(98282)
Evaluate arccos(98282).
0.11065722
0.11065722
Find the Angle Between the Vectors (5,5) , (4,5)

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