The problem asks us to find the volume of a parallelepiped defined by three edge vectors. The edge vectors are given as $3i - 4j + 2k$, $-i + 2j + k$, and $3i - 2j + 5k$.

1. Problem Description

The problem asks us to find the volume of a parallelepiped defined by three edge vectors. The edge vectors are given as $3i - 4j + 2k$, $-i + 2j + k$, and $3i - 2j + 5k$.

2. Solution Steps

The volume of a parallelepiped formed by three vectors $a$, $b$, and $c$ is given by the absolute value of the scalar triple product, which is the determinant of the matrix formed by the components of the vectors:

$V = |a \cdot (b \times c)| = |det(a, b, c)|$.

Let the vectors be:

$a = 3i - 4j + 2k = <3, -4, 2>$

$b = -i + 2j + k = <-1, 2, 1>$

$c = 3i - 2j + 5k = <3, -2, 5>$

We compute the determinant of the matrix formed by these vectors:

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

$V = 3 \begin{vmatrix} 2 & 1 \\ -2 & 5 \end{vmatrix} - (-4) \begin{vmatrix} -1 & 1 \\ 3 & 5 \end{vmatrix} + 2 \begin{vmatrix} -1 & 2 \\ 3 & -2 \end{vmatrix}$

$V = 3(2(5) - 1(-2)) + 4((-1)(5) - 1(3)) + 2((-1)(-2) - 2(3))$

$V = 3(10 + 2) + 4(-5 - 3) + 2(2 - 6)$

$V = 3(12) + 4(-8) + 2(-4)$

$V = 36 - 32 - 8$

$V = 36 - 40$

$V = -4$

The volume is the absolute value of the determinant, so $V = |-4| = 4$.

3. Final Answer

The volume of the parallelepiped is 4.