Question #41878

Let $\vec{w}_1 = 2\hat{i} - 3\hat{j} + \hat{k}$, $\vec{w}_2 = \hat{i} + 2\hat{j} - 2\hat{k}$, and $\vec{w}_3 = -2\hat{i} + \hat{j} + 3\hat{k}$. For $j \in \{1, 2, 3\}$, let $\hat{e}_j$ denote the standard basis vectors $\hat{i}, \hat{j}, \hat{k}$ respectively. Let $\vec{p}_{k,j}$ be the vector projection of $\vec{w}_k$ onto $\hat{e}_j$. Define the vectors $\vec{u}_j = \sum_{k=1}^3 |\vec{p}_{k,j}| \vec{w}_k$ for $j=1, 2, 3$. If $V$ is the volume of the parallelepiped determined by $\vec{u}_1, \vec{u}_2, \vec{u}_3$, find the value of $V$.