`u = 2i - 3j, v = i - 2j` Find the angle theta between the vectors.

Expert Answers

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You need to use the formula of dot product to find the angle between two vectors, `u = u_x*i + u_y*j, v = v_x*i + v_y*j` , such that:

`u*v = |u|*|v|*cos(theta)`

The angle between the vectors u and v is theta.

`cos theta = (u*v)/(|u|*|v|)`

First, you need to evaluate the product of the vectors u and v, such that:

`u*v = u_x*v_x + u_y*v_y`

`u*v = 2*1 + (-3)*(-2)`

`u*v = 8`

You need to evaluate the magnitudes |u| and |v|, such that:

`|u|= sqrt(u_x^2 + u_y^2) => |u|= sqrt(2^2 + (-3)^2) =>|u|= sqrt 13 `

`|v|= sqrt(v_x^2 + v_y^2) => |v|= sqrt(1^2 + (-2)^2) => |v|= sqrt 5`

`cos theta = (8)/(sqrt(13*5)) => cos theta = (8)/(sqrt 65)`

Hence, the cosine of the angle between the vectors u and v is `cos theta = (8)/(sqrt 65)` , so, `theta ~~ 7º 15' 8.089".`

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