\(\begin{aligned}
& \overrightarrow{\mathrm{p}}=\lambda(\overrightarrow{\mathrm{u}} \times \overrightarrow{\mathrm{v}})+\mu(\overrightarrow{\mathrm{v}} \times \overrightarrow{\mathrm{w}})+\mathrm{v}(\overrightarrow{\mathrm{w}} \times \overrightarrow{\mathrm{u}}) \\
& \Rightarrow \vec{p} \cdot \vec{w}=\lambda(\vec{u} \times \vec{v}) \cdot \vec{w}+\mu(\vec{v} \times \vec{w}) \cdot \vec{w} \\
& +v(\vec{w} \times \vec{u}) \cdot \vec{w} \\
& =\lambda[\overrightarrow{\mathrm{u}} \overrightarrow{\mathrm{v}} \overrightarrow{\mathrm{w}}]+0+0=\frac{\lambda}{5} \Rightarrow \lambda=5(\overrightarrow{\mathrm{p}} \cdot \overrightarrow{\mathrm{w}}) \\
&
\end{aligned}\)
Similarly, \(\mu=5(\vec{p} \cdot \vec{u})\) and \(v=5(\vec{p} \cdot \vec{v})\)
\(\begin{aligned}
& \therefore \lambda+\mu+\mathrm{v}=5(\overrightarrow{\mathrm{p}} \cdot \overrightarrow{\mathrm{w}})+5(\overrightarrow{\mathrm{p}} \cdot \overrightarrow{\mathrm{u}})+5(\overrightarrow{\mathrm{p}} \cdot \overrightarrow{\mathrm{v}}) \\
& 5 \overrightarrow{\mathrm{p}} \cdot(\overrightarrow{\mathrm{u}}+\overrightarrow{\mathrm{v}}+\overrightarrow{\mathrm{w}})
\end{aligned}\)
Hence, \(\lambda+\mu+v\) depends on the vectors