(-3,-3) , (-1,2)

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⋅-1-3⋅2

Simplify.

Remove parentheses.

-3⋅-1-3⋅2

Simplify each term.

Multiply -3 by -1.

3-3⋅2

Multiply -3 by 2.

3-6

3-6

Subtract 6 from 3.

-3

-3

-3

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

Add 9 and 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

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+(2)2

Simplify.

Raise -1 to the power of 2.

1+(2)2

Raise 2 to the power of 2.

1+4

Add 1 and 4.

5

5

5

Substitute the values into the equation for the angle between the vectors.

θ=arccos(-3(32)⋅(5))

Cancel the common factor of -3 and 3.

Factor 3 out of -3.

arccos(3⋅-132⋅5)

Cancel the common factors.

Factor 3 out of 32⋅5.

arccos(3⋅-13(2⋅5))

Cancel the common factor.

arccos(3⋅-13(2⋅5))

Rewrite the expression.

arccos(-12⋅5)

arccos(-12⋅5)

arccos(-12⋅5)

Simplify the denominator.

Combine using the product rule for radicals.

arccos(-12⋅5)

Multiply 2 by 5.

arccos(-110)

arccos(-110)

Multiply -110 by 1010.

arccos(-110⋅1010)

Combine and simplify the denominator.

Multiply -110 and 1010.

arccos(-101010)

Raise 10 to the power of 1.

arccos(-1010110)

Raise 10 to the power of 1.

arccos(-10101101)

Use the power rule aman=am+n to combine exponents.

arccos(-10101+1)

Add 1 and 1.

arccos(-10102)

Rewrite 102 as 10.

Use axn=axn to rewrite 10 as 1012.

arccos(-10(1012)2)

Apply the power rule and multiply exponents, (am)n=amn.

arccos(-101012⋅2)

Combine 12 and 2.

arccos(-101022)

Cancel the common factor of 2.

Cancel the common factor.

arccos(-101022)

Divide 1 by 1.

arccos(-10101)

arccos(-10101)

Evaluate the exponent.

arccos(-1010)

arccos(-1010)

arccos(-1010)

Move the negative in front of the fraction.

arccos(-1010)

Evaluate arccos(-1010).

1.89254688

1.89254688

Find the Angle Between the Vectors (-3,-3) , (-1,2)