# Find the Angle Between the Vectors (3,3) , (5,7)

(3,3) , (5,7)
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.
3⋅5+3⋅7
Simplify.
Remove parentheses.
3⋅5+3⋅7
Simplify each term.
Multiply 3 by 5.
15+3⋅7
Multiply 3 by 7.
15+21
15+21
36
36
36
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.
(3)2+(3)2
Simplify.
Raise 3 to the power of 2.
9+(3)2
Raise 3 to the power of 2.
9+9
18
Rewrite 18 as 32⋅2.
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.
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.
(5)2+(7)2
Simplify.
Raise 5 to the power of 2.
25+(7)2
Raise 7 to the power of 2.
25+49
74
74
74
Substitute the values into the equation for the angle between the vectors.
θ=arccos(36(32)⋅(74))
Simplify.
Cancel the common factor of 36 and 3.
Factor 3 out of 36.
arccos(3⋅1232⋅74)
Cancel the common factors.
Factor 3 out of 32⋅74.
arccos(3⋅123(2⋅74))
Cancel the common factor.
arccos(3⋅123(2⋅74))
Rewrite the expression.
arccos(122⋅74)
arccos(122⋅74)
arccos(122⋅74)
Simplify the denominator.
Combine using the product rule for radicals.
arccos(122⋅74)
Multiply 2 by 74.
arccos(12148)
arccos(12148)
Simplify the denominator.
Rewrite 148 as 22⋅37.
Factor 4 out of 148.
arccos(124(37))
Rewrite 4 as 22.
arccos(1222⋅37)
arccos(1222⋅37)
Pull terms out from under the radical.
arccos(12237)
arccos(12237)
Cancel the common factor of 12 and 2.
Factor 2 out of 12.
arccos(2⋅6237)
Cancel the common factors.
Factor 2 out of 237.
arccos(2⋅62(37))
Cancel the common factor.
arccos(2⋅6237)
Rewrite the expression.
arccos(637)
arccos(637)
arccos(637)
Multiply 637 by 3737.
arccos(637⋅3737)
Combine and simplify the denominator.
Multiply 637 and 3737.
arccos(6373737)
Raise 37 to the power of 1.
arccos(63737137)
Raise 37 to the power of 1.
arccos(637371371)
Use the power rule aman=am+n to combine exponents.
arccos(637371+1)
arccos(637372)
Rewrite 372 as 37.
Use axn=axn to rewrite 37 as 3712.
arccos(637(3712)2)
Apply the power rule and multiply exponents, (am)n=amn.
arccos(6373712⋅2)
Combine 12 and 2.
arccos(6373722)
Cancel the common factor of 2.
Cancel the common factor.
arccos(6373722)
Divide 1 by 1.
arccos(637371)
arccos(637371)
Evaluate the exponent.
arccos(63737)
arccos(63737)
arccos(63737)
Evaluate arccos(63737).
0.16514867
0.16514867
Find the Angle Between the Vectors (3,3) , (5,7)