`u = cos(pi/3)i + sin(pi/3)j, v = cos((3pi)/4)i + sin((3pi)/4)j` Find the angle theta between the vectors.

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 = cos(pi/3)*cos(3pi/4) + sin(pi/3)*sin(3pi/4)`

`sin (3pi/4) = sin(pi/2+pi/4) =...

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 = cos(pi/3)*cos(3pi/4) + sin(pi/3)*sin(3pi/4)`

`sin (3pi/4) = sin(pi/2+pi/4) = cos(pi/4) = sqrt2/2`

`cos(3pi/4) = cos(pi/2+pi/4) =-sin(pi/4) = -sqrt2/2`

`u*v = -cos(pi/3)*sin(pi/4) + sin(pi/3)*sin(pi/4)`

`u*v = sqrt2/2*(sqrt3/2 - 1/2)`

`u*v = cos(3pi/4 - pi/3) = cos(5pi/12) = (sqrt2*(sqrt3 - 1))/4`

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

`|u|= sqrt(cos^2(pi/3) + sin^2(pi/3)) => |u|= sqrt(1) =>|u|= 1`

`|v|= sqrt(cos^2(3pi/4) + sin^2(3pi/4)) => |v|= sqrt(1) =>|v|= 1`

`cos theta = (cos(5pi/12))/(1*1) => cos theta =cos(5pi/12) => theta =5pi/12`

Hence, the cosine of the angle between the vectors u and v is `cos theta =cos(5pi/12)` , so, `theta =5pi/12.`