Question 10 Find the volume of the parallelepiped determined by the vectors \( \vec{a}=\langle 4,5,-1\rangle, \vec{b}=\langle 0,1,3\rangle \), \( \vec{c}=\langle 2,4,1\rangle \). Volume \( =\square \) pubic-units Submit Question Question 11

To find the volume of the parallelepiped formed by the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\), you can use the scalar triple product. This is calculated as the absolute value of the determinant of the matrix formed by these vectors. If you put the vectors in a 3x3 matrix: \[ \begin{bmatrix} 4 & 0 & 2 \\ 5 & 1 & 4 \\ -1 & 3 & 1 \\ \end{bmatrix} \] The volume formula is \(|\vec{a} \cdot (\vec{b} \times \vec{c})|\). Calculating the determinant will give you the volume of the parallelepiped, which will yield a fun insight into the spatial relationship of these vectors! One fascinating aspect of the parallelepiped is its resemblance to the three-dimensional extension of a rectangle (or box). Visually, you can think of it as a 3D structure that uses the given vectors as edges originating from a common point. It helps us visualize relationships between vectors in space, showing how linear combinations of these vectors can define a shape – crucial in fields like physics and engineering for modeling various structures!