Answer: \(\mathbf{r}_P=40t\mathbf{i}+(-30+42t)\mathbf{j}\), \(\mathbf{r}_Q=(-10+72t)\mathbf{i}+(18+21t)\mathbf{j}\); they do not collide.

(a) The direction vector for \(P\) is \(20\mathbf{i}+21\mathbf{j}\), whose magnitude is

\(\sqrt{20^2+21^2}=29.\)

Since the speed is \(58\), the velocity vector is

\(\frac{58}{29}(20\mathbf{i}+21\mathbf{j})=40\mathbf{i}+42\mathbf{j}.\)

The initial position vector is \(-30\mathbf{j}\), so

\(\mathbf{r}_P=40t\mathbf{i}+(-30+42t)\mathbf{j}.\)

(b) Since \(\tan\alpha=\frac{7}{24}\), a direction vector for \(Q\) is \(24\mathbf{i}+7\mathbf{j}\). Its magnitude is \(25\).

With speed \(75\), the velocity vector is

\(\frac{75}{25}(24\mathbf{i}+7\mathbf{j})=72\mathbf{i}+21\mathbf{j}.\)

The initial position vector is \(-10\mathbf{i}+18\mathbf{j}\), so

\(\mathbf{r}_Q=(-10+72t)\mathbf{i}+(18+21t)\mathbf{j}.\)

(c) If the particles collide, both components must be equal at the same time.

From the \(\mathbf{i}\)-components,

\(40t=-10+72t,\)

so

\(t=\frac{5}{16}.\)

From the \(\mathbf{j}\)-components,

\(-30+42t=18+21t,\)

so

\(t=\frac{16}{7}.\)

The two values of \(t\) are not the same, so \(P\) and \(Q\) do not collide.