The altitude of the parallelopiped, whose coterminus edges are the vectors $\bar{a}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overline{\mathrm{b}}=2 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}-\hat{\mathrm{k}}, \overline{\mathrm{c}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+3 \hat{\mathrm{k}}$, where $\bar{a}, \overline{\mathrm{~b}}$ are the sides of the base of parallelopiped, is

Given the coterminous edges of the parallelepiped:

$\bar{a} = \hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$

$\bar{b} = 2\hat{\mathrm{i}}+4\hat{\mathrm{j}}-\hat{\mathrm{k}}$

$\bar{c} = \hat{\mathrm{i}}+\hat{\mathrm{j}}+3\hat{\mathrm{k}}$

The volume is the absolute value of the scalar triple product:

$V = |(\bar{a} \times \bar{b}) \cdot \bar{c}| = |\begin{vmatrix} 1 & 1 & 1 \\ 2 & 4 & -1 \\ 1 & 1 & 3 \end{vmatrix}| = |1(12+1) - 1(6+1) + 1(2-4)| = |13 - 7 - 2| = 4$

The base area is the magnitude of the cross product $\bar{a} \times \bar{b}$:

$\bar{a} \times \bar{b} = \begin{vmatrix} \hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\ 1 & 1 & 1 \\ 2 & 4 & -1 \end{vmatrix} = -5\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$

$|\bar{a} \times \bar{b}| = \sqrt{(-5)^2 + 3^2 + 2^2} = \sqrt{38}$

The altitude is given by $h = \frac{V}{|\bar{a} \times \bar{b}|} = \frac{4}{\sqrt{38}}$

Final altitude: $\frac{4}{\sqrt{38}}$