(1,9) , (-3,9)

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.

1⋅-3+9⋅9

Simplify.

Remove parentheses.

1⋅-3+9⋅9

Simplify each term.

Multiply -3 by 1.

-3+9⋅9

Multiply 9 by 9.

-3+81

-3+81

Add -3 and 81.

78

78

78

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+(9)2

Simplify.

One to any power is one.

1+(9)2

Raise 9 to the power of 2.

1+81

Add 1 and 81.

82

82

82

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+(9)2

Simplify.

Raise -3 to the power of 2.

9+(9)2

Raise 9 to the power of 2.

9+81

Add 9 and 81.

90

Rewrite 90 as 32⋅10.

Factor 9 out of 90.

9(10)

Rewrite 9 as 32.

32⋅10

32⋅10

Pull terms out from under the radical.

310

310

310

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

θ=arccos(78(82)⋅(310))

Cancel the common factor of 78 and 3.

Factor 3 out of 78.

arccos(3⋅2682⋅(310))

Cancel the common factors.

Factor 3 out of 82⋅(310).

arccos(3⋅263(82⋅(10)))

Cancel the common factor.

arccos(3⋅263(82⋅(10)))

Rewrite the expression.

arccos(2682⋅(10))

arccos(2682⋅(10))

arccos(2682⋅(10))

Simplify the denominator.

Combine using the product rule for radicals.

arccos(2682⋅10)

Multiply 82 by 10.

arccos(26820)

arccos(26820)

Simplify the denominator.

Rewrite 820 as 22⋅205.

Factor 4 out of 820.

arccos(264(205))

Rewrite 4 as 22.

arccos(2622⋅205)

arccos(2622⋅205)

Pull terms out from under the radical.

arccos(262205)

arccos(262205)

Cancel the common factor of 26 and 2.

Factor 2 out of 26.

arccos(2⋅132205)

Cancel the common factors.

Factor 2 out of 2205.

arccos(2⋅132(205))

Cancel the common factor.

arccos(2⋅132205)

Rewrite the expression.

arccos(13205)

arccos(13205)

arccos(13205)

Multiply 13205 by 205205.

arccos(13205⋅205205)

Combine and simplify the denominator.

Multiply 13205 and 205205.

arccos(13205205205)

Raise 205 to the power of 1.

arccos(132052051205)

Raise 205 to the power of 1.

arccos(1320520512051)

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

arccos(132052051+1)

Add 1 and 1.

arccos(132052052)

Rewrite 2052 as 205.

Use axn=axn to rewrite 205 as 20512.

arccos(13205(20512)2)

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

arccos(1320520512⋅2)

Combine 12 and 2.

arccos(1320520522)

Cancel the common factor of 2.

Cancel the common factor.

arccos(1320520522)

Divide 1 by 1.

arccos(132052051)

arccos(132052051)

Evaluate the exponent.

arccos(13205205)

arccos(13205205)

arccos(13205205)

Evaluate arccos(13205205).

0.43240777

0.43240777

Find the Angle Between the Vectors (1,9) , (-3,9)