Write the vector $\mathbf{v}$ in the form $a \mathbf{i}+b \mathbf{j}$, where $\mathbf{v}$ has the given magnitude and direction angle. Give your answer in exact form. \[ \|\mathrm{v}\|=16, \theta=150^{\circ} \]

Step 1 ：Given the magnitude of the vector \(\|\mathrm{v}\|=16\) and the direction angle \(\theta=150^\circ\).

Step 2 ：The vector \(\mathbf{v}\) can be represented in the form \(a \mathbf{i}+b \mathbf{j}\), where \(a\) and \(b\) are the x and y components of the vector respectively.

Step 3 ：We can find the x and y components using the formulas: \(a = \|\mathrm{v}\| \cos(\theta)\) and \(b = \|\mathrm{v}\| \sin(\theta)\).

Step 4 ：We need to convert the angle from degrees to radians before using these formulas.

Step 5 ：Using the given magnitude and direction angle, we find that \(a = -13.86\) and \(b = 8\).

Step 6 ：The x and y components of the vector are approximately -13.86 and 8 respectively. These are the coefficients of \(\mathbf{i}\) and \(\mathbf{j}\) in the vector representation.

Step 7 ：Final Answer: The vector \(\mathbf{v}\) in the form \(a \mathbf{i}+b \mathbf{j}\) is approximately \(-13.86 \mathbf{i}+8 \mathbf{j}\). In exact form, the vector is \(\boxed{-13.86 \mathbf{i}+8 \mathbf{j}}\).