Scalar Triple Product
An architecture firm in Querétaro is designing a skylight shaped like a pyramid with a triangular base (a tetrahedron). You are given the coordinates of its four vertices and asked how much space it encloses. With the dot product you measure angles, and with the cross product, areas. To measure volumes you combine the two: that is the scalar triple product, which at heart is a $3\times3$ determinant. At the end you will also use it to study lines in space.
Definition: the 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…
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.
Absolute value: volume of the parallelepiped
Height and angle with a plane
Since $V=\text{(area of the base)}\cdot\text{height}$, if the base is the parallelogram of $\vec v$ and $\vec w$:
$$h=\frac{\left|[\vec u,\vec v,\vec w]\right|}{\left|\vec v\times\vec w\right|}$$This $h$ is the length of the projection of $\vec u$ onto $\vec v\times\vec w$. The angle $\theta$ that $\vec u$ makes with the plane of $\vec v$ and $\vec w$ satisfies $\sin\theta = \dfrac{h}{|\vec u|}$.
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 $\vec{b} = (0, 5, 0)$,…
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…
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 $\vec{w} = (1, 2, 5)$.
- a) Calculate the total volume of the hangar.
- 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?
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…
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 $\vec{w} = (k, 1, k+1)$.
- 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 $k$.
- b) Deduce whether there is any value of $k$ that makes the component flat (zero volume).
Coplanarity: when the volume is zero
When a datum is a parameter, for example the coordinate $a$ of a point, set up $[\vec{PQ},\vec{PR},\vec{PS}]=0$ and solve: you get the relation the parameters must satisfy for the points to be coplanar.
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…
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…
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.
Consider four points in space: $P(1, 1, 1)$, $Q(2, 3, a)$, $R(k, 1, 2)$, and $S(0, 0, 3)$. a) Determine the relation that $a$ and $k$ must satisfy for the four points to…
Consider four points in space: $P(1, 1, 1)$, $Q(2, 3, a)$, $R(k, 1, 2)$, and $S(0, 0, 3)$.
- a) Determine the relation that $a$ and $k$ must satisfy for the four points to be coplanar.
- 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.
Properties of the scalar triple product
With these properties you can prove identities without using components: expand by linearity, eliminate the terms with repeated vectors, and reorder (each swap changes the sign). Geometrically, if a result is $2[\vec a,\vec b,\vec c]$, the new parallelepiped has twice the volume.
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,…
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 $\vec{z} = \vec{w} - \vec{v}$.
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…
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 $\vec{w}$.
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:…
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:
$$[\vec{a} + \vec{b}, \vec{b} + \vec{c}, \vec{c} + \vec{a}] = 2 \cdot [\vec{a}, \vec{b}, \vec{c}]$$Interpret the result geometrically by comparing the volume of the original parallelepiped with that formed by the sums of its direction vectors.
Volume of a tetrahedron
Return to the beginning: for the Querétaro skylight you compute $\vec{AB}$, $\vec{AC}$, $\vec{AD}$, their determinant, the absolute value, and divide by $6$.
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…
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 $D(0, 0, z)$.…
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 $Z$.
Consider the tetrahedron with vertices $A(1, 0, 1)$, $B(k, 1, 1)$, $C(1, k, 0)$, and $D(1, 1, k)$. a) Determine the values of the real parameter $k$ for which the volume…
Consider the tetrahedron with vertices $A(1, 0, 1)$, $B(k, 1, 1)$, $C(1, k, 0)$, and $D(1, 1, k)$.
- a) Determine the values of the real parameter $k$ for which the volume of the tetrahedron is $2$ cubic units.
- b) For $k = 2$, find the height of the tetrahedron from vertex $D$ onto the face formed by vertices $A$, $B$, and $C$.
Equations of a line in space
A line is determined by a point $P(x_0,y_0,z_0)$ and a direction vector $\vec d=(a,b,c)$. A point $X$ is on the line if $\vec{PX}$ is parallel to $\vec d$.
Find a point and a direction vector of the line $r$ whose symmetric equation is: $$r: \frac{x-2}{3} = \frac{2y+4}{-2} = 5-z$$
Find a point and a direction vector of the line $r$ whose symmetric equation is:
$$r: \frac{x-2}{3} = \frac{2y+4}{-2} = 5-z$$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…
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: $$\begin{cases} x = 1 + 2\lambda \\ y = -3\lambda \\ z = 2 + \lambda \end{cases}$$ Determine, giving reasons,…
Given the line $r$ expressed in parametric equations:
$$\begin{cases} x = 1 + 2\lambda \\ y = -3\lambda \\ z = 2 + \lambda \end{cases}$$Determine, giving reasons, whether the points $A(3, -3, 3)$ and $B(1, 0, 5)$ lie on this line.
From the implicit form to other forms
Each plane $Ax+By+Cz+D=0$ has normal vector $\vec n=(A,B,C)$. The line that is the intersection of two planes is perpendicular to both normals, so its direction vector is
$$\vec d=\vec n_1\times\vec n_2$$Another way to solve the system: take one variable as a parameter ($z=\lambda$) and solve for the other two in terms of it.
Express in parametric form the line $s$ defined as the intersection of the following planes (implicit equations): $$s: \begin{cases} x - 2y + z = 3 \\ 2x + y - z = 1…
Express in parametric form the line $s$ defined as the intersection of the following planes (implicit equations):
$$s: \begin{cases} x - 2y + z = 3 \\ 2x + y - z = 1 \end{cases}$$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 $\pi_2: 2x - y + z = 0$. a) Express the…
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 $\pi_2: 2x - y + z = 0$.
- a) Express the equation of the trajectory of the laser in symmetric form.
- 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…
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 $5$.
Points and distances on a line
A generic point of a line is obtained from its parametric form: $X(x_0+\lambda a,\ y_0+\lambda b,\ z_0+\lambda c)$. With it you can solve several problems.
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Parallel through a point: same direction $\vec d$ and the new point.
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Midpoint of $B$ and $C$: $M=\frac{B+C}{2}$. The median from $A$ is the line through $A$ and $M$.
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Point at a given distance $L$ from $A$ in the direction $\vec d$: $A\pm L\dfrac{\vec d}{|\vec d|}$.
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Distance from $Q$ to the line: the minimum distance occurs when $\vec{QX}\cdot\vec d=0$.
Given the line $r: \frac{x - 3}{2} = \frac{y + 1}{1} = \frac{z}{2}$: a) Find the equation of a line $s$ passing through the origin of coordinates and parallel to $r$. b)…
Given the line $r: \frac{x - 3}{2} = \frac{y + 1}{1} = \frac{z}{2}$:
- a) Find the equation of a line $s$ passing through the origin of coordinates and parallel to $r$.
- 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 $C(2, 0, 1)$. a) Find the equation of the line containing the median from vertex $A$. b)…
Consider a triangle in space whose vertices are $A(1, 1, 0)$, $B(0, 2, 1)$ and $C(2, 0, 1)$.
- a) Find the equation of the line containing the median from vertex $A$.
- b) Find a point $D$ on this line such that the vector $\vec{AD}$ has magnitude $\sqrt{6}$.
Relative position of two lines
Two lines $r$ (point $A$, direction $\vec d_r$) and $s$ (point $B$, direction $\vec d_s$) can be coincident, parallel, intersecting (they meet), or skew (neither parallel nor intersecting: they do not lie in one plane).
If the lines depend on a parameter $a$, set the determinant equal to zero to find when they intersect. In that case, for the intersection point equate the parametric forms ($A+\lambda\vec d_r=B+\mu\vec d_s$), and the plane containing them has normal $\vec d_r\times\vec d_s$. If you are asked for them to be coincident, the point of one must lie on the other and the directions must be proportional: this gives equations for the parameters.
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…
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.
Study, depending on the real parameter $a$, the relative position of the lines $r$ and $s$: $r: \frac{x-1}{a} = \frac{y-2}{-1} = \frac{z}{2}$…
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.
Two aircraft follow straight-line trajectories given by: $r: \begin{cases} x = 1 + \lambda \\ y = 2 - \lambda \\ z = 3 + 2\lambda \end{cases}$…
Two aircraft follow straight-line trajectories given by:
$r: \begin{cases} x = 1 + \lambda \\ y = 2 - \lambda \\ z = 3 + 2\lambda \end{cases}$
$s: \frac{x-a}{1} = \frac{y}{1} = \frac{z-1}{2}$
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Calculate the value of $a$ for the aircraft trajectories to intersect at a point.
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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.
Planes from lines
A plane parallel to two lines has normal $\vec d_r\times\vec d_s$. A plane that contains a line and is perpendicular to another direction is obtained from the point of the line and the directions of both. A line contained in a plane and perpendicular to $r$ has direction $\vec n\times\vec d_r$ (perpendicular to $\vec n$, because it lies in the plane, and to $\vec d_r$).
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…
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 $s$: $r: \frac{x}{2} = \frac{y-1}{1} = \frac{z}{1}$…
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).
Common perpendicular to two skew lines
If $r$ and $s$ are skew, there is a unique line $t$ that meets both and is perpendicular to both: the common perpendicular.
Volume and skew lines
Let $\vec v=\vec{PQ}$ with $P\in r$ and $Q\in s$. If you change $P$ or $Q$, $\vec v$ changes by a multiple of $\vec d_r$ or of $\vec d_s$, and by the property of adding multiples, $[\vec d_r,\vec d_s,\vec v]$ does not change. That volume is the same for any choice of points, and its value, divided by the area $|\vec d_r\times\vec d_s|$, gives the distance between the lines.
Given the lines $r$ and $s$ defined by the following equations: $r: \begin{cases} x + y - z = 1 \\ 2x - z = 3 \end{cases}$…
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.
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…
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?