Question

What is the angle θ between vectors A⃗ and B⃗ if A⃗ =4ı^−4ȷ^ and B⃗ =−5ı^+7ȷ^?

Answer 1

The characteristics of the scalar product allows to find the angle between the two vectors is:

• The angle θ = 170º

The scalar product is the product between two vectors whose result is a scalar.

            A . B = |A|  |B| cos θ

Where A and B are the vectors, |A| and |B| are the modules of the vectors and θ at the angle between them.

The vector is given in Cartesian coordinates and the unit vectors in these coordinates are perpendicular.

            i.i = j.j = 1

            i.j = 0

            A . B = (4 i - 4j). * -5 i + 7j)

            A . B = - 4 5 - 4 7

            A. B = -48

We look for the modulus of each vector.

           |A| = $\sqrt{x^2 +y^2 }$

           |A| = $\sqrt{4^2 + 4^2}$  

           |A| = 4 √2

          |B| = $\sqrt{5^2 +7^2}$

          |B| = 8.60

We substitute.

            -48 = 4√2  8.60  cos θ

            -48 = 48.66 cos θ

            θ = cos⁻¹   $\frac{-48}{48.664}$  

            θ = 170º

In conclusion using the dot product we can find the angle between the two vectors is:

• the angle θ = 170º

Learn more about the scalar product here:  brainly.com/question/1550649