Find the Angle Between the Vectors (10,5) , (23,-4)

Math
(10,5) , (23,-4)
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.
10⋅23+5⋅-4
Simplify.
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Remove parentheses.
10⋅23+5⋅-4
Simplify each term.
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Multiply 10 by 23.
230+5⋅-4
Multiply 5 by -4.
230-20
230-20
Subtract 20 from 230.
210
210
210
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.
(10)2+(5)2
Simplify.
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Raise 10 to the power of 2.
100+(5)2
Raise 5 to the power of 2.
100+25
Add 100 and 25.
125
Rewrite 125 as 52⋅5.
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Factor 25 out of 125.
25(5)
Rewrite 25 as 52.
52⋅5
52⋅5
Pull terms out from under the radical.
55
55
55
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.
(23)2+(-4)2
Simplify.
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Raise 23 to the power of 2.
529+(-4)2
Raise -4 to the power of 2.
529+16
Add 529 and 16.
545
545
545
Substitute the values into the equation for the angle between the vectors.
θ=arccos(210(55)⋅(545))
Simplify.
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Cancel the common factor of 210 and 5.
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Factor 5 out of 210.
arccos(5⋅4255⋅545)
Cancel the common factors.
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Factor 5 out of 55⋅545.
arccos(5⋅425(5⋅545))
Cancel the common factor.
arccos(5⋅425(5⋅545))
Rewrite the expression.
arccos(425⋅545)
arccos(425⋅545)
arccos(425⋅545)
Simplify the denominator.
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Combine using the product rule for radicals.
arccos(425⋅545)
Multiply 5 by 545.
arccos(422725)
arccos(422725)
Simplify the denominator.
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Rewrite 2725 as 52⋅109.
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Factor 25 out of 2725.
arccos(4225(109))
Rewrite 25 as 52.
arccos(4252⋅109)
arccos(4252⋅109)
Pull terms out from under the radical.
arccos(425109)
arccos(425109)
Multiply 425109 by 109109.
arccos(425109⋅109109)
Combine and simplify the denominator.
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Multiply 425109 and 109109.
arccos(421095109109)
Move 109.
arccos(421095(109109))
Raise 109 to the power of 1.
arccos(421095(1091109))
Raise 109 to the power of 1.
arccos(421095(10911091))
Use the power rule aman=am+n to combine exponents.
arccos(4210951091+1)
Add 1 and 1.
arccos(4210951092)
Rewrite 1092 as 109.
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Use axn=axn to rewrite 109 as 10912.
arccos(421095(10912)2)
Apply the power rule and multiply exponents, (am)n=amn.
arccos(421095⋅10912⋅2)
Combine 12 and 2.
arccos(421095⋅10922)
Cancel the common factor of 2.
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Cancel the common factor.
arccos(421095⋅10922)
Divide 1 by 1.
arccos(421095⋅1091)
arccos(421095⋅1091)
Evaluate the exponent.
arccos(421095⋅109)
arccos(421095⋅109)
arccos(421095⋅109)
Multiply 5 by 109.
arccos(42109545)
Evaluate arccos(42109545).
0.63583842
0.63583842
Find the Angle Between the Vectors (10,5) , (23,-4)

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