Suppose $\bar{u}=\langle-3,1,4\rangle$ and $\bar{v}=\langle-1,3,3\rangle$. Then:
\[
\begin{array}{r}
\bar{u}+\bar{v}= \\
\bar{u}-\bar{v}= \\
\bar{v}-\bar{u}= \\
3 \bar{u}= \\
-\frac{1}{7} \bar{v}= \\
5 \bar{u}-8 \bar{v}=
\end{array}
\]

Step 1 ：Given vectors \(\bar{u} = \langle -3, 1, 4 \rangle\) and \(\bar{v} = \langle -1, 3, 3 \rangle\)

Step 2 ：Adding or subtracting vectors is done by adding or subtracting the corresponding components of the vectors. Scaling the vectors by a scalar is done by multiplying each component of the vector by the scalar.

Step 3 ：Calculate \(\bar{u} + \bar{v}\) by adding the corresponding components of \(\bar{u}\) and \(\bar{v}\), which gives \(\langle -4, 4, 7 \rangle\)

Step 4 ：Calculate \(\bar{u} - \bar{v}\) by subtracting the corresponding components of \(\bar{v}\) from \(\bar{u}\), which gives \(\langle -2, -2, 1 \rangle\)

Step 5 ：Calculate \(\bar{v} - \bar{u}\) by subtracting the corresponding components of \(\bar{u}\) from \(\bar{v}\), which gives \(\langle 2, 2, -1 \rangle\)

Step 6 ：Calculate \(3\bar{u}\) by multiplying each component of \(\bar{u}\) by 3, which gives \(\langle -9, 3, 12 \rangle\)

Step 7 ：Calculate \(-\frac{1}{7}\bar{v}\) by multiplying each component of \(\bar{v}\) by -\frac{1}{7}, which gives \(\langle 0.14285714, -0.42857143, -0.42857143 \rangle\)

Step 8 ：Calculate \(5\bar{u} - 8\bar{v}\) by first scaling each vector by the given scalar, and then subtracting the corresponding components of the vectors, which gives \(\langle -7, -19, -4 \rangle\)

Step 9 ：Final Answer: \(\bar{u} + \bar{v} = \boxed{\langle -4, 4, 7 \rangle}\), \(\bar{u} - \bar{v} = \boxed{\langle -2, -2, 1 \rangle}\), \(\bar{v} - \bar{u} = \boxed{\langle 2, 2, -1 \rangle}\), \(3\bar{u} = \boxed{\langle -9, 3, 12 \rangle}\), \(-\frac{1}{7}\bar{v} = \boxed{\langle 0.14285714, -0.42857143, -0.42857143 \rangle}\), \(5\bar{u} - 8\bar{v} = \boxed{\langle -7, -19, -4 \rangle}\)