Find the Angle Between the Vectors (-3,8) , (2,-2)

(-3,8) , (2,-2)
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⋅2+8⋅-2
Simplify.
Remove parentheses.
-3⋅2+8⋅-2
Simplify each term.
Multiply -3 by 2.
-6+8⋅-2
Multiply 8 by -2.
-6-16
-6-16
Subtract 16 from -6.
-22
-22
-22
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+(8)2
Simplify.
Raise -3 to the power of 2.
9+(8)2
Raise 8 to the power of 2.
9+64
73
73
73
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+(-2)2
Simplify.
Raise 2 to the power of 2.
4+(-2)2
Raise -2 to the power of 2.
4+4
8
Rewrite 8 as 22⋅2.
Factor 4 out of 8.
4(2)
Rewrite 4 as 22.
22⋅2
22⋅2
Pull terms out from under the radical.
22
22
22
Substitute the values into the equation for the angle between the vectors.
θ=arccos(-22(73)⋅(22))
Simplify.
Cancel the common factor of -22 and 2.
Factor 2 out of -22.
arccos(2⋅-1173⋅(22))
Cancel the common factors.
Factor 2 out of 73⋅(22).
arccos(2⋅-112(73⋅(2)))
Cancel the common factor.
arccos(2⋅-112(73⋅(2)))
Rewrite the expression.
arccos(-1173⋅(2))
arccos(-1173⋅(2))
arccos(-1173⋅(2))
Simplify the denominator.
Combine using the product rule for radicals.
arccos(-1173⋅2)
Multiply 73 by 2.
arccos(-11146)
arccos(-11146)
Multiply -11146 by 146146.
arccos(-11146⋅146146)
Combine and simplify the denominator.
Multiply -11146 and 146146.
arccos(-11146146146)
Raise 146 to the power of 1.
arccos(-111461461146)
Raise 146 to the power of 1.
arccos(-1114614611461)
Use the power rule aman=am+n to combine exponents.
arccos(-111461461+1)
arccos(-111461462)
Rewrite 1462 as 146.
Use axn=axn to rewrite 146 as 14612.
arccos(-11146(14612)2)
Apply the power rule and multiply exponents, (am)n=amn.
arccos(-1114614612⋅2)
Combine 12 and 2.
arccos(-1114614622)
Cancel the common factor of 2.
Cancel the common factor.
arccos(-1114614622)
Divide 1 by 1.
arccos(-111461461)
arccos(-111461461)
Evaluate the exponent.
arccos(-11146146)
arccos(-11146146)
arccos(-11146146)
Move the negative in front of the fraction.
arccos(-(11)146146)
Evaluate arccos(-11146146).
2.71496516
2.71496516
Find the Angle Between the Vectors (-3,8) , (2,-2)