# 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
-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
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
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)
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)