# 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
Add -4 and 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
Add 16 and 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
Add 1 and 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)
Add 1 and 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
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