Question

Find the volume of the parallelepiped determined by the vectors \( \left[\begin{array}{l}4 \\ 0 \\ 3\end{array}\right],\left[\begin{array}{c}-4 \\ -1 \\ 0\end{array}\right],\left[\begin{array}{l}0 \\

Answer 1

To find the volume of the parallelepiped determined by the given vectors, we can use the scalar triple product formula. The volume can be calculated by finding the magnitude of the cross product of two of the vectors and then taking the dot product of the result with the third vector. The scalar triple product represents the volume of a parallelepiped.

Step-by-step explanation:

To find the volume of the parallelepiped determined by the given vectors, we need to compute the scalar triple product of the three vectors.

The scalar triple product can be found using the formula:

(B x C) · A = |B x C| * |A| * cos(θ)

where B x C is the cross product of B and C, |B x C| is the magnitude of B x C, |A| is the magnitude of A, and θ is the angle between B x C and A.

1. Calculate B x C using the cross product formula.
2. Calculate the magnitude of B x C.
3. Calculate the magnitude of A.
4. Find the angle θ using the dot product formula: cos(θ) = (B x C) · A / (|B x C| * |A|).
5. Finally, compute the volume of the parallelepiped using the formula: volume = |B x C| * |A| * cos(θ).