Answer
$v_A=5.16ft/s$, $39.8^{\circ}$
Work Step by Step
The required velocity can be determined as follows:
$v_{C/B}=v_C-v_B$
$\implies v_B=v_C+\omega\times r_{B/C}$
We plug in the known values to obtain:
$3\hat i=-4\hat i+(-\omega \hat k)(2.25\hat j)$
$\implies \omega=3.111 rad/s$
Now according to the relative motion between Point $A$ and $C$
$v_A=v_C+v_{A/C}$
$\implies v_A=v_C+\omega \times r_{A/C}$
$\implies v_{Ax}\hat i+v_{Ay}\hat j=-4\hat i +(-3.111\hat k)\times (-1.5 cos 45\hat i)+(1.5+1.5cos45)\hat j$
$\implies v_{Ax}\hat i+v_{Ay}\hat j=(3.967\hat i+3.3 \hat j)ft/s$
Comparing $i$ and $j$ components, we obtain:
$v_{Ax}=3.967ft/s$ and $v_{Ay}=3.3ft/s$
The magnitude of $v_A$ can be determined as
$v_A=\sqrt{v_{Ax}^2+v_{Ay}^2}$
We plug in the known values to obtain:
$v_A=\sqrt{(3.967)^2+(3.3)^2}$
$\implies v_A=5.16ft/s$
The direction is given as
$\theta=tan^{-1}(\frac{v_{Ay}}{v_{Ax}})$
$\implies \theta=tan^{-1}(\frac{3.967}{3.3})=39.8^{\circ}$
