Find the Angle Between the Vectors (3,3) , (5,7)

Math
(3,3) , (5,7)
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.
3⋅5+3⋅7
Simplify.
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Remove parentheses.
3⋅5+3⋅7
Simplify each term.
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Multiply 3 by 5.
15+3⋅7
Multiply 3 by 7.
15+21
15+21
Add 15 and 21.
36
36
36
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.
(3)2+(3)2
Simplify.
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Raise 3 to the power of 2.
9+(3)2
Raise 3 to the power of 2.
9+9
Add 9 and 9.
18
Rewrite 18 as 32⋅2.
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Factor 9 out of 18.
9(2)
Rewrite 9 as 32.
32⋅2
32⋅2
Pull terms out from under the radical.
32
32
32
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.
(5)2+(7)2
Simplify.
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Raise 5 to the power of 2.
25+(7)2
Raise 7 to the power of 2.
25+49
Add 25 and 49.
74
74
74
Substitute the values into the equation for the angle between the vectors.
θ=arccos(36(32)⋅(74))
Simplify.
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Cancel the common factor of 36 and 3.
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Factor 3 out of 36.
arccos(3⋅1232⋅74)
Cancel the common factors.
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Factor 3 out of 32⋅74.
arccos(3⋅123(2⋅74))
Cancel the common factor.
arccos(3⋅123(2⋅74))
Rewrite the expression.
arccos(122⋅74)
arccos(122⋅74)
arccos(122⋅74)
Simplify the denominator.
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Combine using the product rule for radicals.
arccos(122⋅74)
Multiply 2 by 74.
arccos(12148)
arccos(12148)
Simplify the denominator.
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Rewrite 148 as 22⋅37.
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Factor 4 out of 148.
arccos(124(37))
Rewrite 4 as 22.
arccos(1222⋅37)
arccos(1222⋅37)
Pull terms out from under the radical.
arccos(12237)
arccos(12237)
Cancel the common factor of 12 and 2.
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Factor 2 out of 12.
arccos(2⋅6237)
Cancel the common factors.
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Factor 2 out of 237.
arccos(2⋅62(37))
Cancel the common factor.
arccos(2⋅6237)
Rewrite the expression.
arccos(637)
arccos(637)
arccos(637)
Multiply 637 by 3737.
arccos(637⋅3737)
Combine and simplify the denominator.
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Multiply 637 and 3737.
arccos(6373737)
Raise 37 to the power of 1.
arccos(63737137)
Raise 37 to the power of 1.
arccos(637371371)
Use the power rule aman=am+n to combine exponents.
arccos(637371+1)
Add 1 and 1.
arccos(637372)
Rewrite 372 as 37.
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Use axn=axn to rewrite 37 as 3712.
arccos(637(3712)2)
Apply the power rule and multiply exponents, (am)n=amn.
arccos(6373712⋅2)
Combine 12 and 2.
arccos(6373722)
Cancel the common factor of 2.
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Cancel the common factor.
arccos(6373722)
Divide 1 by 1.
arccos(637371)
arccos(637371)
Evaluate the exponent.
arccos(63737)
arccos(63737)
arccos(63737)
Evaluate arccos(63737).
0.16514867
0.16514867
Find the Angle Between the Vectors (3,3) , (5,7)

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