$D$ is on side $\overline {AB}$ of $\triangle ABC.$ We know $\triangle ABC \sim \triangle ACD$ and we know all the marked side lengths. [asy] size(5cm); pair A, B,C, D; A=(0,0); B=(4,0); C=2*dir(48); D=foot(C, A,B) + (-0.3,0); draw(C--A--B--C--D); label("$A$",A, SW); label("$B$",B, SE); label("$C$",C, N); label("$D$",D, S); label("$3$", A--D, S); label("$6$", A--C, W); [/asy] What is the length $AB?$