(5,5) , (4,5)

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.

5⋅4+5⋅5

Simplify.

Remove parentheses.

5⋅4+5⋅5

Simplify each term.

Multiply 5 by 4.

20+5⋅5

Multiply 5 by 5.

20+25

20+25

Add 20 and 25.

45

45

45

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

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+(5)2

Simplify.

Raise 4 to the power of 2.

16+(5)2

Raise 5 to the power of 2.

16+25

Add 16 and 25.

41

41

41

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

θ=arccos(45(52)⋅(41))

Cancel the common factor of 45 and 5.

Factor 5 out of 45.

arccos(5⋅952⋅41)

Cancel the common factors.

Factor 5 out of 52⋅41.

arccos(5⋅95(2⋅41))

Cancel the common factor.

arccos(5⋅95(2⋅41))

Rewrite the expression.

arccos(92⋅41)

arccos(92⋅41)

arccos(92⋅41)

Simplify the denominator.

Combine using the product rule for radicals.

arccos(92⋅41)

Multiply 2 by 41.

arccos(982)

arccos(982)

Multiply 982 by 8282.

arccos(982⋅8282)

Combine and simplify the denominator.

Multiply 982 and 8282.

arccos(9828282)

Raise 82 to the power of 1.

arccos(98282182)

Raise 82 to the power of 1.

arccos(982821821)

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

arccos(982821+1)

Add 1 and 1.

arccos(982822)

Rewrite 822 as 82.

Use axn=axn to rewrite 82 as 8212.

arccos(982(8212)2)

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

arccos(9828212⋅2)

Combine 12 and 2.

arccos(9828222)

Cancel the common factor of 2.

Cancel the common factor.

arccos(9828222)

Divide 1 by 1.

arccos(982821)

arccos(982821)

Evaluate the exponent.

arccos(98282)

arccos(98282)

arccos(98282)

Evaluate arccos(98282).

0.11065722

0.11065722

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