# Find the Angle Between the Vectors (10,7) , (-2,1)

(10,7) , (-2,1)
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.
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⋅-2+7⋅1
Simplify.
Remove parentheses.
10⋅-2+7⋅1
Simplify each term.
Multiply 10 by -2.
-20+7⋅1
Multiply 7 by 1.
-20+7
-20+7
Add -20 and 7.
-13
-13
-13
Find the magnitude of u.
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+(7)2
Simplify.
Raise 10 to the power of 2.
100+(7)2
Raise 7 to the power of 2.
100+49
Add 100 and 49.
149
149
149
Find the magnitude of v.
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+(1)2
Simplify.
Raise -2 to the power of 2.
4+(1)2
One to any power is one.
4+1
Add 4 and 1.
5
5
5
Substitute the values into the equation for the angle between the vectors.
θ=arccos(-13(149)⋅(5))
Simplify.
Simplify the denominator.
Combine using the product rule for radicals.
arccos(-13149⋅5)
Multiply 149 by 5.
arccos(-13745)
arccos(-13745)
Multiply -13745 by 745745.
arccos(-13745⋅745745)
Combine and simplify the denominator.
Multiply -13745 and 745745.
arccos(-13745745745)
Raise 745 to the power of 1.
arccos(-137457451745)
Raise 745 to the power of 1.
arccos(-1374574517451)
Use the power rule aman=am+n to combine exponents.
arccos(-137457451+1)
Add 1 and 1.
arccos(-137457452)
Rewrite 7452 as 745.
Use axn=axn to rewrite 745 as 74512.
arccos(-13745(74512)2)
Apply the power rule and multiply exponents, (am)n=amn.
arccos(-1374574512⋅2)
Combine 12 and 2.
arccos(-1374574522)
Cancel the common factor of 2.
Cancel the common factor.
arccos(-1374574522)
Divide 1 by 1.
arccos(-137457451)
arccos(-137457451)
Evaluate the exponent.
arccos(-13745745)
arccos(-13745745)
arccos(-13745745)
Move the negative in front of the fraction.
arccos(-(13)745745)
Evaluate arccos(-13745745).
2.06721908
2.06721908
Find the Angle Between the Vectors (10,7) , (-2,1)

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