Scalar triple product

Given the vectors $\vec{u} = (2, -1, 3)$, $\vec{v} = (0, 4, 1)$, and $\vec{w} = (1, 1, -2)$, calculate the value of the scalar triple product $[\vec{u}, \vec{v}, \vec{w}]$ using the definition based on the determinant of their components.

An engineering student is designing a component using CAD software. The base of the component is defined by the vectors $\vec{a} = (3, 0, 0)$ and $\vec{b} = (0, 5, 0)$, while the third vector defining the depth is $\vec{c} = (1, 2, 4)$. Calculate the volume of the parallelepiped formed by these three vectors.

Consider a storage hangar in the shape of a parallelepiped where three of its edges meeting at a vertex are the vectors $\vec{u} = (3, 0, 0)$, $\vec{v} = (0, 4, 0)$, and $\vec{w} = (1, 2, 5)$.

1. a) Calculate the total volume of the hangar.
2. b) If the hangar is divided by an inclined plane passing through the endpoints of the three previous vectors to create a triangular mezzanine (tetrahedron with the origin), what fraction of the total volume of the hangar would this mezzanine occupy?

Check, giving reasons, whether the vectors $\vec{a} = (1, 2, -1)$, $\vec{b} = (3, 0, 2)$, and $\vec{c} = (4, 2, 1)$ are coplanar. To do this, calculate their scalar triple product and justify your answer based on the result obtained.

Check whether the points $P(1, 2, -1)$, $Q(3, 0, 2)$, $R(2, 1, 4)$, and $S(5, -2, 7)$ are coplanar using the scalar triple product. If they are not, calculate the volume of the tetrahedron that has these four points as vertices.

Given three linearly independent vectors $\vec{u}$, $\vec{v}$, and $\vec{w}$ such that their scalar triple product is $[\vec{u}, \vec{v}, \vec{w}] = 4$, calculate, giving reasons and applying the properties of determinants, the value of the scalar triple product of the vectors $\vec{x} = 2\vec{u}$, $\vec{y} = \vec{u} + \vec{v}$, and $\vec{z} = \vec{w} - \vec{v}$.

Given the vectors $\vec{u} = (1, 0, 1)$ and $\vec{v} = (2, 1, -1)$, find a third vector $\vec{w}$ of the form $(0, 0, z)$ such that the volume of the parallelepiped defined by the three is exactly 5 cubic units.

An architectural firm designs a skylight in the shape of a tetrahedron whose vertices are the points $A(1, 1, 1)$, $B(3, 2, 0)$, $C(2, 4, -1)$, and $D(0, 0, z)$. Calculate the value of the z-coordinate $z$ knowing that the volume of the skylight is $5$ cubic units and that vertex $D$ lies on the negative semi-axis of the $Z$.