Problem $1.$

In the mechanism shown below, slider $C$ slides along bar $AB.$ Bar CD is $0\mathrm{.}5$ m in length. At the instant shown, link $AB$ is rotating with angular velocity ${}_{AB}={}_3^{}=3ra\frac{d}{s}(CCW),$ link $CD$ is horizontal, and distance $AC=0\mathrm{.}75m$ as shown.

Using the complex number approach, answer the following:

$($a$)$ Write the loop closure equation for the velocity vector expression.

$($b$)$ Specify the zero and nonzero terms in the velocity expression.

$($c$)$ Write the real and imaginary equations of the velocity expression

$($d$)$ Solve the real and imaginary equations of the velocity expression to determine

$,r_1$ and $r_4($use Pythagorean theorem for this part$)$

the angular velocity of link $CD[$Magnitude and direction$]$

the relative velocity of point $C$ with respect to Magnitude and direction$]$