Introduction to Three Dimensional Geometry

The distance of point $P(x, y, z)$ from the origin is:

In the three-dimensional coordinate system, the three coordinate planes divide the space into how many parts?

A point $R$ is in the $XZ$-plane. Its $y$-coordinate is:

The point $(-3, -4, 5)$ lies in which octant?

A point $P$ is located in the three-dimensional coordinate system such that its perpendicular distances from the $XY$-plane, $YZ$-plane, and $XZ$-plane are 3, 4, and 5 units respectively. If the point lies in the octant where $x < 0$, $y > 0$, and $z < 0$, what are the coordinates of point $P$?

The mid-point of the line segment joining $P(2, 3, 4)$ and $Q(4, 1, -2)$ is:

What is the perpendicular distance of the point $(2, 3, 4)$ from the $x$-axis?

Find the length of the diagonal of a rectangular parallelepiped (box) whose faces are parallel to the coordinate planes and whose opposite vertices are $(2, 3, 5)$ and $(5, 7, 17)$.

A point $P$ moves such that the sum of its distances from the $XY$-plane and the $YZ$-plane is always equal to its distance from the $ZX$-plane. If $P$ is in the first octant, what is the equation of the plane on which $P$ lies?

In a 3D coordinate system, the image of point $P(1, 2, 3)$ in the $XY$-plane is $Q$, and the image of $Q$ in the $YZ$-plane is $R$. What are the coordinates of $R$?

If the point $(x, y, z)$ is on the $z$-axis, then:

The distance of point $P(2, 3, 4)$ from the origin is:

What are the coordinates of a point $P$ which is the midpoint of the line segment joining $A(2, -3, 4)$ and $B(8, -1, 2)$?

In the $xy$-plane, the $z$-coordinate of every point is:

In the rectangular coordinate system shown below, a point $P$ is located in the three-dimensional space such that its perpendicular distance from the $XY$-plane is 5 units, from the $YZ$-plane is 3 units, and from the $XZ$-plane is 4 units. If point $P$ lies in the octant where $x < 0$, $y > 0$, and $z < 0$, determine its coordinates $(x, y, z)$.

Determine the octant in which the point $(-3, 1, -2)$ lies.

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 given by:

Find the equation of the set of points $P$ such that the sum of the squares of its distances from the points $A(3, 4, 5)$ and $B(-1, 3, -7)$ is equal to $k^2 = 50$. What is the constant term in the resulting equation $2x^2 + 2y^2 + 2z^2 - 4x - 14y + 4z + C = 0$?

A point $R$ with $x$-coordinate $4$ lies on the line segment joining the points $P(2, -3, 4)$ and $Q(8, 0, 10)$. Find the coordinates of $R$.

Find the coordinates of the point which is equidistant from the four points $O(0, 0, 0)$, $A(a, 0, 0)$, $B(0, b, 0)$, and $C(0, 0, c)$. This point is the circumcenter of the tetrahedron $OABC$.

Find the distance from point $(1, 2, 3)$ to the x-axis.

Find the distance between $P(1, -3, 4)$ and $Q(4, 1, 2)$.

Find the distance between $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$.

What is the distance between $A(2, 3, 5)$ and $B(-1, 3, 2)$?

A point $P(2, -1, 3)$ is located in 3D space. Find the distance between point $P$ and its reflection $P'$ in the $yz$-plane.

If the distance between the points $(3, a, -1)$ and $(1, -3, 5)$ is $10$ units, find the possible values of $a$.

What is the perimeter of a triangle with vertices $(0, 1, 2)$, $(1, 2, 0)$ and $(2, 0, 1)$?

A surveyor is measuring a terrain. He marks three points $A(k, 1, 5)$, $B(1, 0, 3)$, and $C(7, 2, 1)$. If these points are vertices of an isosceles triangle with $AB = BC$, find the value of $k$ and the distance $AC$.

The coordinates of three vertices of a tetrahedron are $O(0,0,0)$, $A(4,0,0)$, and $B(0,4,0)$. The fourth vertex $C$ lies on the line $x=y=z$. If the volume of the tetrahedron is $\frac{32}{3}$ cubic units, and $z > 0$, calculate the distance of $C$ from the centroid of the base $\triangle OAB$.

A laser beam originates from $P(1, -1, 1)$ and hits a mirror at $M$ on the plane $z=0$ (the $xy$-plane) and reflects to pass through $Q(3, 3, 3)$. If the path $PM + MQ$ is the shortest possible, find the coordinates of $M$ and the total distance traveled by the beam.

Find the coordinates of the centroid of the triangle whose vertices are $(0, 0, 0)$, $(a, 0, 0)$, and $(0, b, 0)$.

Find the ratio in which the segment joining $(1, 2, 3)$ and $(-3, 4, -5)$ is divided by the $z$-axis.

If the distance between points $(x, 0, 0)$ and $(1, 2, 3)$ is 7 units, find $x$.

If the vertices of a triangle are $A(5, 4, 6)$, $B(1, -1, 3)$, and $C(4, 3, 2)$, what is the z-coordinate of its centroid?

The coordinates of the point which divides the segment joining $A(2, 3, 4)$ and $B(3, 5, 6)$ in the ratio $1:-2$ (external division $1:2$) are:

If the points $A(3, 2, 0)$, $B(5, 3, 2)$, and $C(-9, 6, -3)$ are vertices of a triangle, find the length of the median from $A$ to $BC$.

The midpoint of the line segment joining $(a, 2b, 4)$ and $(6, 4, 2c)$ is $(2, 8, 10)$. Find $a, b, c$.

Two vertices of a triangle are $A(3, 5, -4)$ and $B(-1, 1, 2)$. If the centroid of the triangle is $G(2, 3, 2)$, find the coordinates of the third vertex $C$.

The midpoints of the sides of a triangle are $(1, 5, -1)$, $(0, 4, -2)$, and $(2, 3, 4)$. Find the coordinates of the centroid of the triangle.

A point $C(-1, 2, 1)$ divides the line segment joining $A(1, -2, 5)$ and $B$ in the ratio $2:3$ internally. Find the coordinates of the point $B$.