We are given three vectors $\vec{OA} = 3\hat{i} - \hat{j}$, $\vec{OB} = \hat{j} + 2\hat{k}$, and $\vec{OC} = \hat{i} + 5\hat{j} + 4\hat{k}$ representing adjacent sides of a parallelepiped. The problem asks us to find the volume of this parallelepiped.

1. Problem Description

We are given three vectors $\vec{OA} = 3\hat{i} - \hat{j}$, $\vec{OB} = \hat{j} + 2\hat{k}$, and $\vec{OC} = \hat{i} + 5\hat{j} + 4\hat{k}$ representing adjacent sides of a parallelepiped. The problem asks us to find the volume of this parallelepiped.

2. Solution Steps

The volume of a parallelepiped with adjacent sides defined by vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ is given by the absolute value of the scalar triple product:

$V = |\vec{a} \cdot (\vec{b} \times \vec{c})|$

This can also be expressed as the absolute value of the determinant of the matrix formed by the components of the vectors:

$V = \left| \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix} \right|$

In our case, the vectors are $\vec{OA} = <3, -1, 0>$, $\vec{OB} = <0, 1, 2>$, and $\vec{OC} = <1, 5, 4>$.

Thus, the volume of the parallelepiped is:

$V = \left| \begin{vmatrix} 3 & -1 & 0 \\ 0 & 1 & 2 \\ 1 & 5 & 4 \end{vmatrix} \right|$

We can compute the determinant as follows:

$V = |3(1 \cdot 4 - 2 \cdot 5) - (-1)(0 \cdot 4 - 2 \cdot 1) + 0(0 \cdot 5 - 1 \cdot 1)|$

$V = |3(4 - 10) + 1(0 - 2) + 0|$

$V = |3(-6) - 2|$

$V = |-18 - 2|$

$V = |-20|$

$V = 20$

3. Final Answer

The volume of the parallelepiped is 20.