Find the Angle Between the Vectors (1,9) , (-3,9)

Math
(1,9) , (-3,9)
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.
1⋅-3+9⋅9
Simplify.
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Remove parentheses.
1⋅-3+9⋅9
Simplify each term.
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Multiply -3 by 1.
-3+9⋅9
Multiply 9 by 9.
-3+81
-3+81
Add -3 and 81.
78
78
78
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.
(1)2+(9)2
Simplify.
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One to any power is one.
1+(9)2
Raise 9 to the power of 2.
1+81
Add 1 and 81.
82
82
82
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.
(-3)2+(9)2
Simplify.
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Raise -3 to the power of 2.
9+(9)2
Raise 9 to the power of 2.
9+81
Add 9 and 81.
90
Rewrite 90 as 32⋅10.
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Factor 9 out of 90.
9(10)
Rewrite 9 as 32.
32⋅10
32⋅10
Pull terms out from under the radical.
310
310
310
Substitute the values into the equation for the angle between the vectors.
θ=arccos(78(82)⋅(310))
Simplify.
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Cancel the common factor of 78 and 3.
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Factor 3 out of 78.
arccos(3⋅2682⋅(310))
Cancel the common factors.
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Factor 3 out of 82⋅(310).
arccos(3⋅263(82⋅(10)))
Cancel the common factor.
arccos(3⋅263(82⋅(10)))
Rewrite the expression.
arccos(2682⋅(10))
arccos(2682⋅(10))
arccos(2682⋅(10))
Simplify the denominator.
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Combine using the product rule for radicals.
arccos(2682⋅10)
Multiply 82 by 10.
arccos(26820)
arccos(26820)
Simplify the denominator.
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Rewrite 820 as 22⋅205.
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Factor 4 out of 820.
arccos(264(205))
Rewrite 4 as 22.
arccos(2622⋅205)
arccos(2622⋅205)
Pull terms out from under the radical.
arccos(262205)
arccos(262205)
Cancel the common factor of 26 and 2.
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Factor 2 out of 26.
arccos(2⋅132205)
Cancel the common factors.
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Factor 2 out of 2205.
arccos(2⋅132(205))
Cancel the common factor.
arccos(2⋅132205)
Rewrite the expression.
arccos(13205)
arccos(13205)
arccos(13205)
Multiply 13205 by 205205.
arccos(13205⋅205205)
Combine and simplify the denominator.
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Multiply 13205 and 205205.
arccos(13205205205)
Raise 205 to the power of 1.
arccos(132052051205)
Raise 205 to the power of 1.
arccos(1320520512051)
Use the power rule aman=am+n to combine exponents.
arccos(132052051+1)
Add 1 and 1.
arccos(132052052)
Rewrite 2052 as 205.
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Use axn=axn to rewrite 205 as 20512.
arccos(13205(20512)2)
Apply the power rule and multiply exponents, (am)n=amn.
arccos(1320520512⋅2)
Combine 12 and 2.
arccos(1320520522)
Cancel the common factor of 2.
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Cancel the common factor.
arccos(1320520522)
Divide 1 by 1.
arccos(132052051)
arccos(132052051)
Evaluate the exponent.
arccos(13205205)
arccos(13205205)
arccos(13205205)
Evaluate arccos(13205205).
0.43240777
0.43240777
Find the Angle Between the Vectors (1,9) , (-3,9)

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