Introduction to Three-Dimensional Geometry

In a three-dimensional coordinate system, a point $P$ is located such that it lies specifically on the $XZ$-plane. If the coordinates of point $P$ are given as $(a, b, c)$, what must be the value of the coordinate $b$?

A point is at a distance of $3$ units from the $yz$-plane, $4$ units from the $xz$-plane and $5$ units from the $xy$-plane. Find its coordinates in the first octant.

Which coordinate is zero for every point on the $xz$-plane?

The three coordinate planes divide the entire space into how many parts?

In which octant are all three coordinates negative?

A point moves so that its distance from the origin is always $3$ units. If it also lies on the plane $z=0$, what curve does it trace?

Find the distance between the point $(2, 3, 4)$ and its reflection in the $xy$-plane.

A submarine at $M(2, -3, -4)$ receives a signal from a station. If the station is at the point obtained by rotating $M$ by $180^\circ$ about the $z$-axis, what are the coordinates of the station?

An architect is modeling a pyramid with a square base. The base vertices in the $xy$-plane are $(0,0,0), (2,0,0), (2,2,0)$, and $(0,2,0)$. The apex is at $(1,1,3)$. What is the length of the lateral edge connecting $(0,0,0)$ to the apex?

In a 3D coordinate system, the locus of points whose distance from the $x$-axis is twice its distance from the $yz$-plane is given by which equation?

A point $P(x, y, z)$ is located in the octant where all coordinates are negative. If the point lies on the plane shown in the figure and its distance from the $yz$-plane is $5$ units, its distance from the $xz$-plane is $2$ units, and its distance from the $xy$-plane is $3$ units, what are the coordinates of point $P$?

If a point $P(x, y, z)$ is such that $x=0, y=0, z \neq 0$, then $P$ lies on:

The coordinates of a point $M$ are $(2, 3, 4)$. What is the distance of $M$ from the $z$-axis?

A point $P$ is in the $yz$-plane. What is its $x$-coordinate?

In the rectangular parallelepiped shown, the origin $O$ is at one vertex, and the edges lie along the positive $x$, $y$, and $z$ axes. If the coordinates of the opposite vertex $A$ are $(4, 5, 6)$, find the coordinates of the point $M$ which is the midpoint of the diagonal $OA$.

If the origin is the centroid of the triangle with vertices $A(a, 1, 3)$, $B(-2, b, -5)$ and $C(4, 7, c)$, find the values of $a, b, c$.

A point $C$ lies on the line joining $A(2, 2, -3)$ and $B(5, 6, 9)$. If the $z$-coordinate of $C$ is $3$, find its $x$-coordinate.

Three vertices of a parallelogram $ABCD$ are $A(3, -1, 2)$, $B(1, 2, -4)$, and $C(-1, 1, 2)$. Find the coordinates of the fourth vertex $D$.

The coordinates of the foot of the perpendicular from the point $P(1, 3, 4)$ to the $XY$-plane are $(x, y, z)$. What is the sum $x+y+z$?

A vector field is defined by coordinates. A particle moves from $P(2, 3, 1)$ to $Q(5, 7, 1)$. Find a point $R$ on the line $PQ$ such that $PR = 2 PQ$ (externally).

The centroid of a triangle with vertices $(x_1, y_1, z_1)$, $(x_2, y_2, z_2)$ and $(x_3, y_3, z_3)$ is:

Find the distance between $(1, -1, 3)$ and $(2, 3, -5)$.

Find the distance of the point $P(x, y, z)$ from the origin.

If $L$ is the foot of the perpendicular from $P(3, 4, 5)$ on the $xy$-plane, then the coordinates of $L$ are:

A line segment joins the points $P(2, -3, 1)$ and $Q(-1, 1, 13)$. A point $R$ lies on this segment such that it is equidistant from the $XY$-plane and the $YZ$-plane. If the distance between points $P$ and $Q$ is $d$ units, calculate $d$ and determine which of the following represents the length of the segment $PQ$?

If the distance between the points $(2, 3, z)$ and $(1, -1, 2)$ is $3\sqrt{2}$, find $z$.

Find the length of the medians of the triangle with vertices $A(0, 0, 6)$, $B(0, 4, 0)$ and $C(6, 0, 0)$. What is the length of the median through $A$?

A laser beam starts at $A(1, 2, 3)$ and hits a mirror at $M$ on the $xy$-plane. The beam is reflected to point $B(5, 6, 7)$. If the path length $AM + MB$ is minimized (Law of Reflection), find this minimum distance.

Two points $P$ and $Q$ are given by $(2, 3, -1)$ and $(x, 1, 1)$. If the distance $PQ$ is $\sqrt{21}$ units and $x > 0$, find the coordinates of a point $R$ on the $x$-axis such that $PQR$ forms a right-angled triangle at $Q$.

A sphere passes through the points $(0,0,0)$, $(a,0,0)$, $(0,b,0)$, and $(0,0,c)$. Find the distance from the origin to the center of this sphere.

The ratio in which the point $(5, 4, -6)$ divides the segment joining $P(3, 2, -10)$ and $Q(9, 8, -2)$ is:

If the midpoint of a segment is $(2, 0, 5)$ and one endpoint is $(3, 4, 2)$, what is the other endpoint?

The coordinates of the point that divides the segment joining $(1, 2, 3)$ and $(4, 5, 6)$ in the ratio $1:2$ internally are:

Find the ratio in which the xy-plane divides the line joining $(4, 8, 10)$ and $(6, 10, -8)$.

The ratio in which the point $(x, y, 6)$ divides the join of $A(3, 5, 4)$ and $B(5, 8, 10)$ is:

If the points $A(1, 0, -6)$, $B(3, p, q)$ and $C(5, 9, 6)$ are collinear, find the values of $p$ and $q$.

Find the coordinates of the points which trisect the line segment joining the points $P(4, 2, -6)$ and $Q(10, -16, 6)$.

The midpoints of the sides of a triangle are $(x_1, y_1, z_1)$, $(x_2, y_2, z_2)$, and $(x_3, y_3, z_3)$. The centroid of this triangle is given by $(\bar{x}, \bar{y}, \bar{z})$. Which of the following is true regarding the centroid of the triangle formed by these midpoints?

A cable is stretched between two hooks at $M(1, -2, 3)$ and $N(3, 4, -5)$. A weight is hung at point $W$ such that $MW = \frac{1}{4} MN$. Find the $z$-coordinate of point $W$.

Points $P(3, 2, -4)$, $Q(5, 4, -6)$, and $R(9, 8, -10)$ are collinear. Find the ratio in which $Q$ divides the segment $PR$.