Find the Angle Between the Vectors (1,2.22) , (2,3.32)

Math
(1,2.22) , (2,3.32)
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⋅2+2.22⋅3.32
Simplify.
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Remove parentheses.
1⋅2+2.22⋅3.32
Simplify each term.
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Multiply 2 by 1.
2+2.22⋅3.32
Multiply 2.22 by 3.32.
2+7.3704
2+7.3704
Add 2 and 7.3704.
9.3704
9.3704
9.3704
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+(2.22)2
Simplify.
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One to any power is one.
1+(2.22)2
Raise 2.22 to the power of 2.
1+4.9284
Add 1 and 4.9284.
5.9284
5.9284
5.9284
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.
(2)2+(3.32)2
Simplify.
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Raise 2 to the power of 2.
4+(3.32)2
Raise 3.32 to the power of 2.
4+11.0224
Add 4 and 11.0224.
15.0224
15.0224
15.0224
Substitute the values into the equation for the angle between the vectors.
θ=arccos(9.3704(5.9284)⋅(15.0224))
Simplify.
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Simplify the denominator.
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Combine using the product rule for radicals.
arccos(9.37045.9284⋅15.0224)
Multiply 5.9284 by 15.0224.
arccos(9.370489.05879616)
arccos(9.370489.05879616)
Multiply 9.370489.05879616 by 89.0587961689.05879616.
arccos(9.370489.05879616⋅89.0587961689.05879616)
Combine and simplify the denominator.
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Multiply 9.370489.05879616 and 89.0587961689.05879616.
arccos(9.370489.0587961689.0587961689.05879616)
Raise 89.05879616 to the power of 1.
arccos(9.370489.0587961689.05879616189.05879616)
Raise 89.05879616 to the power of 1.
arccos(9.370489.0587961689.05879616189.058796161)
Use the power rule aman=am+n to combine exponents.
arccos(9.370489.0587961689.058796161+1)
Add 1 and 1.
arccos(9.370489.0587961689.058796162)
Rewrite 89.058796162 as 89.05879616.
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Use axn=axn to rewrite 89.05879616 as 89.0587961612.
arccos(9.370489.05879616(89.0587961612)2)
Apply the power rule and multiply exponents, (am)n=amn.
arccos(9.370489.0587961689.0587961612⋅2)
Combine 12 and 2.
arccos(9.370489.0587961689.0587961622)
Cancel the common factor of 2.
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Cancel the common factor.
arccos(9.370489.0587961689.0587961622)
Divide 1 by 1.
arccos(9.370489.0587961689.058796161)
arccos(9.370489.0587961689.058796161)
Evaluate the exponent.
arccos(9.370489.0587961689.05879616)
arccos(9.370489.0587961689.05879616)
arccos(9.370489.0587961689.05879616)
Multiply 9.3704 by 89.05879616.
arccos(88.4293719289.05879616)
Divide 88.42937192 by 89.05879616.
arccos(0.99293248)
Evaluate arccos(0.99293248).
0.11896095
0.11896095
Find the Angle Between the Vectors (1,2.22) , (2,3.32)

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