find the volume of the parallelepiped with one vertex at the origin and adjacent vertices at (1, 3, 0), (-2, 0, 2),(-1, 3, -1).

This problem aims to find the volume of a parallelepiped, whose one vertex is at the origin (0,0) and the other 3 vertices are given. To solve this problem, it is required to have knowledge of 3-dimensional shapes along with their areas and volumes and to calculate determinants of the 3×3 square matrix.

A parallelepiped is a 3-dimensional shape formed by six individual parallelograms. It is related to a parallelogram the same as a cube is related to a square.To keep things simple, we will construct a 3×3 matrix A, where the column entries are coordinates of the adjacent vertices of the given parallelepiped.$A=\left[\begin {matrix}1&-2&-1\\3& &3\\0&2&-1\\\end {matrix}\right]$The formula to find the volume is a dot product of the base of the parallelogram and its tilted altitude. But in matrix notation, the parallelepiped volume is equal to the absolute value of the determinant of $A$.Volume = $|det(A)|$Adjusting the matrix $A$ in the formula gives us:$volume=\left|\begin{matrix}1&-2&-1\\3&0&3\\0&2&-1\\\end{matrix}\right|$Next, we will solve for $det(A)$. Note that the determinant can only be found in a square matrix such as $A$.We will find the determinant by using co-factor expansion across the first column.$=\left|\begin{matrix}0&3\\2&-1\\\end{matrix}\right|-3\left|\begin{matrix}-2& -1\\2& -1\\ \end {matrix} \right| +0 \left |\begin {matrix} -2 & -1\\ 0 & 3\\ \end {matrix} \right|$
Expanding the first column gives us only 2 entries as $a_13$ is equal to 0, but a complete solution is given here for simplicity.$= [ (0)(-1) – (2)(3) ] + (-3)[ (-2)(-1) – (2)(-1) ]$$= -6 + (-3)[ 2 +2]$$= -6 + (-3)(4)$$= -6 + (-3)(4)$$= -6 – 12$$volume = -18$Therefore, the volume of the given parallelepiped is equal to $18$.
Find the volume of the parallelepiped with one vertex at the origin and adjacent vertices at $(1, 0, -3), (1, 2, 4), (5, 1, 0)$.As the first step, we will construct a $3\times3$ matrix $A$, whose column entries are coordinates of the adjacent vertices of the given parallelepiped.$A = \left [\begin {matrix} 1 & 1 & 5 \\ 0 & 2 & 1\\ -3 & 4 & 0\\ \end {matrix} \right]$The volume of the parallelepiped can be calculated by taking absolute value of determinant of $A$.$Volume = |det(A)|$Adjusting the matrix $A$ in the formula gives us:$volume = \left |\begin {matrix} 1 & 1 & 5 \\ 0 & 2 & 1\\ -3 & 4 & 0\\ \end {matrix} \right|$Next, we will solve for $det(A)$ by using co-factor expansion across the first column.$= \left |\begin {matrix} 2 & 1\\ 4 & 0\\ \end {matrix} \right| -(0) \left |\begin {matrix} 1 & 5\\ 4 & 0\\ \end {matrix} \right| +(-3) \left |\begin {matrix} 1 & 5\\ 2 & 1\\ \end {matrix} \right|$Equation becomes:$v = -4+27$$volume = 23$Thus, the volume of parallelepiped comes out to be $23$.