Distance Formula in 3D

If $A\left(-1,0,0\right), B\left(0,1,0\right), C\left(0,0,-1\right)$ are the vertices of $△ABC. \text{If} R \text{and} r$ denote the circum radius and in radius of $△ABC,$ then which of the following statement (s) is/are correct $?$

Show that the points $A\left(1, 2, 3\right), B\left(-1, -2, -1\right), C\left(2, 3, 2\right)$ and $D\left(4, 7, 6\right)$  are the vertices of a parallelogram $ABCD$, but it is not a rectangle.

Find the equation of set of points $P$ such that $P{A}^2+P{B}^2=2k^2$, where $A$ and $B$ are the points $\left(3, 4, 5\right)$ and $\left(-1, 3, -7\right)$, respectively.

Are the points $A\left(3, 6, 9\right), B\left(10, 20, 30\right)$ and $C\left(25, -41, 5\right)$, the vertices of a right angled triangle?

Show that the points $P\left(-2, 3, 5\right), Q\left(1, 2, 3\right)$ and $R\left(7, 0, -1\right)$ are collinear.

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

A tracking station lies at the origin of a coordinate system with the $x$-axis due east, the $y$-axis due north and the $z$-axis vertically upwards. Two aircraft have coordinates $(20,25,11)\text{ and }(26,31,12)$ relative to the tracking station. The radar at the tracking station has a range of $40km$. Determine whether it will be able to detect both aircraft.

A tracking station lies at the origin of a coordinate system with the $x$-axis due east, the $y$-axis due north and the $z$-axis vertically upwards. Two aircraft have coordinates $(20,25,11)\text{ and }(26,31,12)$ relative to the tracking station. Find the distance between the two aircraft at this time.

Find the distance between the pair of points given below:

$\mathrm{A}(1,-2,10)$ and $\mathrm{B}(-4,0,3)$

Verify whether the four points $\left(5, -1, 1\right), \left(-1, -3, 4\right),$ $\left(7, -4, 7\right)$ and $\left(1, -6, 10\right)$ are the vertices of rhombus or not.

Show that the points $\left(0, 7, 10\right), \left(-1, 6, 6\right)$ and $\left(-4, 9, 6\right)$ form a right angled triangle.

 Find the distance (in units) between the pair of points $P(1, -1, 0)$and $Q (2, 1, 2)$.

If the distance from the origin to $(6, 6,7)$ is $k$ units, then $k=$

How far apart are the points $(2,0,0)$ and $(–3,0,0)$?

If the distance between the points $(a, 2, 1)$ and $(1, –1, 1)$ is  $5$, then $a=$ _____. (write only positive value of $a$)

Find the length and the foot of perpendicular from the point $\left(1,\frac{3}{2},2\right)$ to the plane $2x-2y+4z+5=0$ is of the form of $\sqrt{a}$, then $a=$

Find the foot of the perpendicular from the point $\left(2,3,-8\right)\text{ to the line }\frac{4-x}{2}=\frac{y}{6}=\frac{1-z}{3}$. Also, if the perpendicular distance from the given point to the line is of the form $3\sqrt{a}$, then $a=$

For a map with a scale of 1:25,000, what is the ground distance represented by 4 cm on the map?

If the scale of a map is 1:200,000, what is the ground distance in kilometers between two points that are 3 cm apart on the map?

A map has a scale of 1:25,000. What is the actual ground distance if the distance between two points on the map is 4 cm?