Answer
$a=-8$
Work Step by Step
Step 1: Since vectors $a\textbf {i}+6\textbf {j}$ and $9\textbf {i}+12\textbf {j}$ are perpendicular, their dot product must be zero.
$(a,6)\cdot(9,12)=a(9)+6(12)=9a+72=0$.
Step 2: Now, to find $b$, we must solve the above equation.
Step 3: $9a+72=0$
Step 4: Subtract 72 from both sides: $9a+72-72=0-72$.
Step 5: $9a=-72$
Step 6: Dividing both sides by $9$, $\frac{9a}{9}=\frac{-72}{9}$
Step 7: Simplifying, $a=-8$.
