(10,7) , (-2,1)

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⋅-2+7⋅1

Simplify.

Remove parentheses.

10⋅-2+7⋅1

Simplify each term.

Multiply 10 by -2.

-20+7⋅1

Multiply 7 by 1.

-20+7

-20+7

Add -20 and 7.

-13

-13

-13

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+(7)2

Simplify.

Raise 10 to the power of 2.

100+(7)2

Raise 7 to the power of 2.

100+49

Add 100 and 49.

149

149

149

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.

(-2)2+(1)2

Simplify.

Raise -2 to the power of 2.

4+(1)2

One to any power is one.

4+1

Add 4 and 1.

5

5

5

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

θ=arccos(-13(149)⋅(5))

Simplify the denominator.

Combine using the product rule for radicals.

arccos(-13149⋅5)

Multiply 149 by 5.

arccos(-13745)

arccos(-13745)

Multiply -13745 by 745745.

arccos(-13745⋅745745)

Combine and simplify the denominator.

Multiply -13745 and 745745.

arccos(-13745745745)

Raise 745 to the power of 1.

arccos(-137457451745)

Raise 745 to the power of 1.

arccos(-1374574517451)

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

arccos(-137457451+1)

Add 1 and 1.

arccos(-137457452)

Rewrite 7452 as 745.

Use axn=axn to rewrite 745 as 74512.

arccos(-13745(74512)2)

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

arccos(-1374574512⋅2)

Combine 12 and 2.

arccos(-1374574522)

Cancel the common factor of 2.

Cancel the common factor.

arccos(-1374574522)

Divide 1 by 1.

arccos(-137457451)

arccos(-137457451)

Evaluate the exponent.

arccos(-13745745)

arccos(-13745745)

arccos(-13745745)

Move the negative in front of the fraction.

arccos(-(13)745745)

Evaluate arccos(-13745745).

2.06721908

2.06721908

Find the Angle Between the Vectors (10,7) , (-2,1)