(-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|)

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

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

Add 9 and 64.

73

73

73

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

Add 4 and 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))

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)

Add 1 and 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)