Exercises: 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.

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.

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.

Given the lines $r$ and $s$ defined by the following equations:

$r: \begin{cases} x + y - z = 1 \\ 2x - z = 3 \end{cases}$

$s: \frac{x-1}{2} = \frac{y+1}{1} = \frac{z}{-1}$

Determine the equation of the line $t$ that is the common perpendicular to $r$ and $s$. To do this, first find the direction vectors of both lines and justify, giving reasons, that they are skew lines in space.

Study, depending on the real parameter $a$, the relative position of the lines $r$ and $s$:

$r: \frac{x-1}{a} = \frac{y-2}{-1} = \frac{z}{2}$

$s: \begin{cases} x + 2y = 3 \\ ax + z = 1 \end{cases}$

In the particular case where the lines intersect, calculate the coordinates of the intersection point $P$ and the equation of the plane containing them.

Consider the point $P(1, 0, -1)$ and the lines:

$r: x = y = z$

$s: \begin{cases} x = 3 + t \\ y = 3 + 2t \\ z = 1 + t \end{cases}$

Find the general equation of the plane $\pi$ that passes through point $P$ and is parallel to both lines. Then, determine the equation of a third line $k$ that passes through point $P$, is contained in said plane $\pi$, and is perpendicular to line $r$.

Find the equation of the line $t$ passing through the point $A(1, 2, 1)$ and intersecting the lines $r$ and $s$:

$r: \frac{x}{2} = \frac{y-1}{1} = \frac{z}{1}$

$s: \begin{cases} x + z = 3 \\ y = 2 \end{cases}$

Explain the method used based on the intersection of planes (pencil of planes).

It is known that the line $r$ passes through the point $A(1, m, 0)$ and has direction vector $\vec{u} = (1, 1, a)$. On the other hand, the line $s$ is the intersection of the planes:

$\pi_1: x - y + z = 0$

$\pi_2: 2x + y = 1$

Determine whether there exist values of the parameters $m$ and $a$ such that lines $r$ and $s$ are the same (coincident lines) and, if so, find them.

Two aircraft follow straight-line trajectories given by:

$r: \begin{cases} x = 1 + \lambda \\ y = 2 - \lambda \\ z = 3 + 2\lambda \end{cases}$

$s: \frac{x-a}{1} = \frac{y}{1} = \frac{z-1}{2}$

1. Calculate the value of $a$ for the aircraft trajectories to intersect at a point.

2. For that value of $a$, find the equation of the line $n$ that is perpendicular to the plane formed by the trajectories $r$ and $s$ and passes through the point of collision.

It is desired to install a fiber optic cable connecting point $A(1, 4, -2)$ to point $B(3, -1, 0)$. Find the equation of the line passing through both points in symmetric form.

Given the line $r$ expressed in parametric equations:

Determine, giving reasons, whether the points $A(3, -3, 3)$ and $B(1, 0, 5)$ lie on this line.

Express in parametric form the line $s$ defined as the intersection of the following planes (implicit equations):

Find a point and a direction vector of the line $r$ whose symmetric equation is:

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 $Z$.

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 $\vec{z} = \vec{w} - \vec{v}$.

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?

The trajectory of a laser beam in an entertainment venue is given by the intersection of the planes $\pi_1: x + y - z = 3$ and $\pi_2: 2x - y + z = 0$.

1. a) Express the equation of the trajectory of the laser in symmetric form.
2. b) A safety sensor is located at the point $S(1, 2, 0)$. Determine whether the laser beam will pass exactly through the position of the sensor.

In the design of an architectural structure, a metal beam lies along the line $r$ passing through the point $A(1, 0, -1)$ and parallel to the line $s: \begin{cases} 3x - y + z = 0 \\ x + y = 2 \end{cases}$.

Obtain the equation of the line $r$ in implicit form (as the intersection of two planes) and calculate a point on it whose $z$-coordinate is equal to $5$.

Given the line $r: \frac{x - 3}{2} = \frac{y + 1}{1} = \frac{z}{2}$:

1. a) Find the equation of a line $s$ passing through the origin of coordinates and parallel to $r$.
2. b) Determine the distance separating the point $Q(1, 1, 1)$ from the line $s$ using a general point on it.

Consider a triangle in space whose vertices are $A(1, 1, 0)$, $B(0, 2, 1)$ and $C(2, 0, 1)$.

1. a) Find the equation of the line containing the median from vertex $A$.
2. b) Find a point $D$ on this line such that the vector $\vec{AD}$ has magnitude $\sqrt{6}$.

Consider the tetrahedron with vertices $A(1, 0, 1)$, $B(k, 1, 1)$, $C(1, k, 0)$, and $D(1, 1, k)$.

1. a) Determine the values of the real parameter $k$ for which the volume of the tetrahedron is $2$ cubic units.
2. b) For $k = 2$, find the height of the tetrahedron from vertex $D$ onto the face formed by vertices $A$, $B$, and $C$.

A team of engineers is designing a support component for a satellite in the shape of a parallelepiped. The base is defined by the vectors $\vec{u} = (2, -1, 0)$ and $\vec{v} = (1, 2, 1)$. The third vector defining the component is $\vec{w} = (k, 1, k+1)$.

1. a) Given that the projection of vector $\vec{w}$ onto the cross product $\vec{u} \times \vec{v}$ has a length of $3$ units, determine the value of $k$.
2. b) Deduce whether there is any value of $k$ that makes the component flat (zero volume).

Using the properties of the scalar triple product, prove vectorially that for any three vectors $\vec{a}, \vec{b}, \vec{c} \in \mathbb{R}^3$, the following holds:

Interpret the result geometrically by comparing the volume of the original parallelepiped with that formed by the sums of its direction vectors.

Consider four points in space: $P(1, 1, 1)$, $Q(2, 3, a)$, $R(k, 1, 2)$, and $S(0, 0, 3)$.

1. a) Determine the relation that $a$ and $k$ must satisfy for the four points to be coplanar.
2. b) If we fix $k = 2$, what value must $a$ take so that the volume of the tetrahedron $PQRS$ is minimal? Justify your answer based on the linear dependence of the vectors.

In a modern architectural structure, a column lies along the common perpendicular to lines $r$ and $s$. Line $r$ passes through the origin and has direction $\vec{d_r} = (1, 0, 1)$. Line $s$ passes through $P(0, 1, 0)$ and has direction $\vec{d_s} = (0, 1, 1)$.

Calculate the volume of the parallelepiped defined by the direction vectors $\vec{d_r}$, $\vec{d_s}$, and a third vector $\vec{v}$ joining any point on $r$ to any point on $s$. Does this volume depend on the points chosen on the lines?

Let $\vec{u}$ be a unit vector and $\vec{v}, \vec{w}$ two vectors such that the area of the parallelogram they form is equal to $10$. Given that the scalar triple product is $[\vec{u}, \vec{v}, \vec{w}] = 5$, calculate the angle that vector $\vec{u}$ makes with the plane containing vectors $\vec{v}$ and $\vec{w}$.