Answer
$a$
Work Step by Step
$(*)\quad {\bf u_{1}}$and ${\bf u_{2}}$ are orthogonal $\Rightarrow{\bf u_{1}}\cdot{\bf u_{2}}=0$
$(**) \quad {\bf u_{1}}$and ${\bf u_{2}}$ are unit vectors $\Rightarrow|{\bf u_{1}}|=|{\bf u_{2}}|=1$
Using Properties of the Dot Product (boxed on p. 709):
${\bf v}\cdot{\bf u_{1}}=(a{\bf u_{1}}+b{\bf u_{2}})\cdot{\bf u_{1}}$
$=a{\bf u_{1}} \cdot{\bf u_{1}}+b{\bf u_{2}}\cdot{\bf u_{1}} \qquad$... property 3
$=a({\bf u_{1}} \cdot{\bf u_{1}})+b({\bf u_{2}}\cdot{\bf u_{1}}) \qquad$... property 2
$=a({\bf u_{1}} \cdot{\bf u_{1}})+b(0) \qquad$... $(*)$ from above
$=a\cdot|{\bf u_{1}}|^{2} \qquad$... property $4$
$=a(1) \qquad$... $(**)$ from above
$=a$
