(10,-9) , (13,-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.

10⋅13-9⋅-5

Simplify.

Remove parentheses.

10⋅13-9⋅-5

Simplify each term.

Multiply 10 by 13.

130-9⋅-5

Multiply -9 by -5.

130+45

130+45

Add 130 and 45.

175

175

175

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.

(10)2+(-9)2

Simplify.

Raise 10 to the power of 2.

100+(-9)2

Raise -9 to the power of 2.

100+81

Add 100 and 81.

181

181

181

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.

(13)2+(-5)2

Simplify.

Raise 13 to the power of 2.

169+(-5)2

Raise -5 to the power of 2.

169+25

Add 169 and 25.

194

194

194

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

θ=arccos(175(181)⋅(194))

Simplify the denominator.

Combine using the product rule for radicals.

arccos(175181⋅194)

Multiply 181 by 194.

arccos(17535114)

arccos(17535114)

Multiply 17535114 by 3511435114.

arccos(17535114⋅3511435114)

Combine and simplify the denominator.

Multiply 17535114 and 3511435114.

arccos(175351143511435114)

Raise 35114 to the power of 1.

arccos(1753511435114135114)

Raise 35114 to the power of 1.

arccos(17535114351141351141)

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

arccos(17535114351141+1)

Add 1 and 1.

arccos(17535114351142)

Rewrite 351142 as 35114.

Use axn=axn to rewrite 35114 as 3511412.

arccos(17535114(3511412)2)

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

arccos(175351143511412⋅2)

Combine 12 and 2.

arccos(175351143511422)

Cancel the common factor of 2.

Cancel the common factor.

arccos(175351143511422)

Divide 1 by 1.

arccos(17535114351141)

arccos(17535114351141)

Evaluate the exponent.

arccos(1753511435114)

arccos(1753511435114)

arccos(1753511435114)

Evaluate arccos(1753511435114).

0.36564126

0.36564126

Find the Angle Between the Vectors (10,-9) , (13,-5)