Find the Angle Between the Vectors (-3,-3) , (-1,2)

Math
(-3,-3) , (-1,2)
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⋅-1-3⋅2
Simplify.
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Remove parentheses.
-3⋅-1-3⋅2
Simplify each term.
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Multiply -3 by -1.
3-3⋅2
Multiply -3 by 2.
3-6
3-6
Subtract 6 from 3.
-3
-3
-3
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.
(-1)2+(2)2
Simplify.
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Raise -1 to the power of 2.
1+(2)2
Raise 2 to the power of 2.
1+4
Add 1 and 4.
5
5
5
Substitute the values into the equation for the angle between the vectors.
θ=arccos(-3(32)⋅(5))
Simplify.
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Cancel the common factor of -3 and 3.
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Factor 3 out of -3.
arccos(3⋅-132⋅5)
Cancel the common factors.
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Factor 3 out of 32⋅5.
arccos(3⋅-13(2⋅5))
Cancel the common factor.
arccos(3⋅-13(2⋅5))
Rewrite the expression.
arccos(-12⋅5)
arccos(-12⋅5)
arccos(-12⋅5)
Simplify the denominator.
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Combine using the product rule for radicals.
arccos(-12⋅5)
Multiply 2 by 5.
arccos(-110)
arccos(-110)
Multiply -110 by 1010.
arccos(-110⋅1010)
Combine and simplify the denominator.
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Multiply -110 and 1010.
arccos(-101010)
Raise 10 to the power of 1.
arccos(-1010110)
Raise 10 to the power of 1.
arccos(-10101101)
Use the power rule aman=am+n to combine exponents.
arccos(-10101+1)
Add 1 and 1.
arccos(-10102)
Rewrite 102 as 10.
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Use axn=axn to rewrite 10 as 1012.
arccos(-10(1012)2)
Apply the power rule and multiply exponents, (am)n=amn.
arccos(-101012⋅2)
Combine 12 and 2.
arccos(-101022)
Cancel the common factor of 2.
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Cancel the common factor.
arccos(-101022)
Divide 1 by 1.
arccos(-10101)
arccos(-10101)
Evaluate the exponent.
arccos(-1010)
arccos(-1010)
arccos(-1010)
Move the negative in front of the fraction.
arccos(-1010)
Evaluate arccos(-1010).
1.89254688
1.89254688
Find the Angle Between the Vectors (-3,-3) , (-1,2)

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