(4,-2) , (5,-5)

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.

4⋅5-2⋅-5

Simplify.

Remove parentheses.

4⋅5-2⋅-5

Simplify each term.

Multiply 4 by 5.

20-2⋅-5

Multiply -2 by -5.

20+10

20+10

Add 20 and 10.

30

30

30

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+(-2)2

Simplify.

Raise 4 to the power of 2.

16+(-2)2

Raise -2 to the power of 2.

16+4

Add 16 and 4.

20

Rewrite 20 as 22⋅5.

Factor 4 out of 20.

4(5)

Rewrite 4 as 22.

22⋅5

22⋅5

Pull terms out from under the radical.

25

25

25

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.

(5)2+(-5)2

Simplify.

Raise 5 to the power of 2.

25+(-5)2

Raise -5 to the power of 2.

25+25

Add 25 and 25.

50

Rewrite 50 as 52⋅2.

Factor 25 out of 50.

25(2)

Rewrite 25 as 52.

52⋅2

52⋅2

Pull terms out from under the radical.

52

52

52

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

θ=arccos(30(25)⋅(52))

Reduce the expression by cancelling the common factors.

Cancel the common factor of 30 and 2.

Factor 2 out of 30.

arccos(2⋅1525⋅(52))

Cancel the common factors.

Factor 2 out of 25⋅(52).

arccos(2⋅152(5⋅(52)))

Cancel the common factor.

arccos(2⋅152(5⋅(52)))

Rewrite the expression.

arccos(155⋅(52))

arccos(155⋅(52))

arccos(155⋅(52))

Cancel the common factor of 15 and 5.

Factor 5 out of 15.

arccos(5⋅35⋅(52))

Cancel the common factors.

Factor 5 out of 5⋅(52).

arccos(5⋅35(5⋅(2)))

Cancel the common factor.

arccos(5⋅35(5⋅(2)))

Rewrite the expression.

arccos(35⋅(2))

arccos(35⋅(2))

arccos(35⋅(2))

arccos(35⋅(2))

Simplify the denominator.

Combine using the product rule for radicals.

arccos(35⋅2)

Multiply 5 by 2.

arccos(310)

arccos(310)

Multiply 310 by 1010.

arccos(310⋅1010)

Combine and simplify the denominator.

Multiply 310 and 1010.

arccos(3101010)

Raise 10 to the power of 1.

arccos(31010110)

Raise 10 to the power of 1.

arccos(310101101)

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

arccos(310101+1)

Add 1 and 1.

arccos(310102)

Rewrite 102 as 10.

Use axn=axn to rewrite 10 as 1012.

arccos(310(1012)2)

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

arccos(3101012⋅2)

Combine 12 and 2.

arccos(3101022)

Cancel the common factor of 2.

Cancel the common factor.

arccos(3101022)

Divide 1 by 1.

arccos(310101)

arccos(310101)

Evaluate the exponent.

arccos(31010)

arccos(31010)

arccos(31010)

Evaluate arccos(31010).

0.32175055

0.32175055

Find the Angle Between the Vectors (4,-2) , (5,-5)