# Find the Angle Between the Vectors (4,3) , (-1,6)

(4,3) , (-1,6)
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.
4⋅-1+3⋅6
Simplify.
Remove parentheses.
4⋅-1+3⋅6
Simplify each term.
Multiply 4 by -1.
-4+3⋅6
Multiply 3 by 6.
-4+18
-4+18
14
14
14
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.
(4)2+(3)2
Simplify.
Raise 4 to the power of 2.
16+(3)2
Raise 3 to the power of 2.
16+9
25
Rewrite 25 as 52.
52
Pull terms out from under the radical, assuming positive real numbers.
5
5
5
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.
(-1)2+(6)2
Simplify.
Raise -1 to the power of 2.
1+(6)2
Raise 6 to the power of 2.
1+36
37
37
37
Substitute the values into the equation for the angle between the vectors.
θ=arccos(14(5)⋅(37))
Simplify.
Multiply 145⋅37 by 3737.
arccos(145⋅37⋅3737)
Combine and simplify the denominator.
Multiply 145⋅37 and 3737.
arccos(14375⋅3737)
Move 37.
arccos(14375(3737))
Raise 37 to the power of 1.
arccos(14375(37137))
Raise 37 to the power of 1.
arccos(14375(371371))
Use the power rule aman=am+n to combine exponents.
arccos(14375371+1)
arccos(14375372)
Rewrite 372 as 37.
Use axn=axn to rewrite 37 as 3712.
arccos(14375(3712)2)
Apply the power rule and multiply exponents, (am)n=amn.
arccos(14375⋅3712⋅2)
Combine 12 and 2.
arccos(14375⋅3722)
Cancel the common factor of 2.
Cancel the common factor.
arccos(14375⋅3722)
Divide 1 by 1.
arccos(14375⋅371)
arccos(14375⋅371)
Evaluate the exponent.
arccos(14375⋅37)
arccos(14375⋅37)
arccos(14375⋅37)
Multiply 5 by 37.
arccos(1437185)
Evaluate arccos(1437185).
1.09244389
1.09244389
Find the Angle Between the Vectors (4,3) , (-1,6)