Exercises: Loci in space

Given the points $A(2, -1, 3)$ and $B(4, 3, -1)$:

1. a) Set up the condition $\text{dist}(X, A) = \text{dist}(X, B)$ for a general point $X(x, y, z)$.
2. b) Simplify the resulting equation to obtain the equation of the perpendicular bisecting plane of segment $AB$.
3. c) Check that the midpoint of $AB$ lies on this plane.

1. a) Find the equation of the locus of points in space at a distance of $5$ units from point $C(1, -2, 0)$.
2. b) What surface is it? State its center and radius.
3. c) Determine, without drawing, whether point $P(4, 2, 0)$ belongs to this surface.

Find the locus of points in space that are equidistant from planes $\pi: 2x - y + 2z - 3 = 0$ and $\pi': x + 2y - 2z + 1 = 0$.

1. a) Set up the condition $\text{dist}(X, \pi) = \text{dist}(X, \pi')$ using the formula for the distance from a point to a plane.
2. b) Solve the equation involving absolute values by distinguishing the two possible cases and give the equations of the resulting planes (bisecting planes).
3. c) Check that the two planes obtained are perpendicular to each other.

Given the points $A(1, 0, 2)$ and $B(3, 4, -2)$, we seek the locus of points $X$ in space such that triangle $ABX$ is a right triangle at $X$, that is, segments $AX$ and $BX$ are perpendicular, as illustrated in the figure.

1. a) Express the condition of perpendicularity using the scalar product $\vv{XA} \times \vv{XB} = 0$.
2. b) Expand the condition to obtain an equation in $x$, $y$, $z$ and rewrite it by completing the square.
3. c) Identify the surface obtained: how are its center and radius related to segment $AB$? (It is the 3D spatial version of Thales' theorem.)

1. a) Justify why the distance from a general point $X(x, y, z)$ to the $OZ$-axis is $\sqrt{x^2 + y^2}$.
2. b) Find the equation of the locus of points at a distance of $3$ units from the $OZ$-axis and identify the surface obtained.
3. c) What shape is formed by the intersection of this surface with the plane $z = 2$? And with the plane $y = 0$?

Two sonar beacons are installed at the bottom of a harbor, at points $A(0, 0, 0)$ and $B(6, 2, 4)$ (coordinates in hectometers). A submarine receives the signal from both with the same intensity, meaning it is equidistant from both beacons, as suggested in the figure.

1. a) Set up the condition $\text{dist}(X, A) = \text{dist}(X, B)$ for a general point $X(x, y, z)$ and show that the locus of the submarine's positions is a plane, finding its equation.
2. b) A submarine is at point $P(4, 2, 1)$. Does it receive both signals with the same intensity? Which beacon is it closer to?
3. c) Interpret the plane obtained as a "coverage boundary" between the two beacons.

Three seismic sensors are placed at points $A(0, 0, 0)$, $B(2, 0, 0)$, and $C(0, 2, 0)$. We wish to install an antenna equidistant from all three.

1. a) Find the perpendicular bisecting plane of line segment $AB$.
2. b) Find the perpendicular bisecting plane of line segment $AC$.
3. c) Deduce that the locus of points equidistant from $A$, $B$, and $C$ is a line, and give its parametric equations. What is the orientation of this line with respect to the plane containing the three sensors?

State whether the following statements about loci in space are true or false, justifying each response:

1. a) "The locus of points equidistant from two distinct parallel planes is a single plane, parallel to both and equidistant from each."
2. b) "The locus of points equidistant from two intersecting planes is a single plane."
3. c) "The locus of points equidistant from the three coordinate axes is a single line."