Answer: (a) \(\begin{pmatrix}-7.5\\4\end{pmatrix}\); (b) \(a=1\), \(b=2\).

Answer: (a) \(\begin{pmatrix}-7.5\\4\end{pmatrix}\); (b) \(a=1\), \(b=2\).

(a) The magnitude of \(\begin{pmatrix}15\\-8\end{pmatrix}\) is \(\sqrt{15^2+(-8)^2}=17\). A vector in the opposite direction is \(\begin{pmatrix}-15\\8\end{pmatrix}\). To make its magnitude \(8.5\), multiply by \(\frac{8.5}{17}=\frac12\):

\(\frac12\begin{pmatrix}-15\\8\end{pmatrix}=\begin{pmatrix}-7.5\\4\end{pmatrix}.\)

(b) Equate components:

\(15a+2a+1=6(b+a),\qquad 5b+2=12.\)

The second equation gives \(b=2\). The first simplifies to

\(11a+1=6b.\)

Substitute \(b=2\):

\(11a+1=12,\)

so \(a=1\).