(4,3) , (-1,6)

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⋅-1+3⋅6

Simplify.

Remove parentheses.

4⋅-1+3⋅6

Simplify each term.

Multiply 4 by -1.

-4+3⋅6

Multiply 3 by 6.

-4+18

-4+18

Add -4 and 18.

14

14

14

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+(3)2

Simplify.

Raise 4 to the power of 2.

16+(3)2

Raise 3 to the power of 2.

16+9

Add 16 and 9.

25

Rewrite 25 as 52.

52

Pull terms out from under the radical, assuming positive real numbers.

5

5

5

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+(6)2

Simplify.

Raise -1 to the power of 2.

1+(6)2

Raise 6 to the power of 2.

1+36

Add 1 and 36.

37

37

37

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

θ=arccos(14(5)⋅(37))

Multiply 145⋅37 by 3737.

arccos(145⋅37⋅3737)

Combine and simplify the denominator.

Multiply 145⋅37 and 3737.

arccos(14375⋅3737)

Move 37.

arccos(14375(3737))

Raise 37 to the power of 1.

arccos(14375(37137))

Raise 37 to the power of 1.

arccos(14375(371371))

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

arccos(14375371+1)

Add 1 and 1.

arccos(14375372)

Rewrite 372 as 37.

Use axn=axn to rewrite 37 as 3712.

arccos(14375(3712)2)

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

arccos(14375⋅3712⋅2)

Combine 12 and 2.

arccos(14375⋅3722)

Cancel the common factor of 2.

Cancel the common factor.

arccos(14375⋅3722)

Divide 1 by 1.

arccos(14375⋅371)

arccos(14375⋅371)

Evaluate the exponent.

arccos(14375⋅37)

arccos(14375⋅37)

arccos(14375⋅37)

Multiply 5 by 37.

arccos(1437185)

Evaluate arccos(1437185).

1.09244389

1.09244389

Find the Angle Between the Vectors (4,3) , (-1,6)