Distance Formula in 3D

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 $a=$

A point moves so that its distances from the points $\left(3,4,-2\right)$ and $\left(2, 3, -3\right)$ remains equal. The locus of the point is

If the product of distances of the point $\left(1, 1, 1\right)$ from the origin and the plane $x-y+z+k=0$ be $5$, then $k=$

The co-ordinates of points $A,B,C,D$ are $\left(a,\mathrm{ }2,\mathrm{ }1\right),\mathrm{ }\left(1,\mathrm{ }-1,\mathrm{ }1\right),\mathrm{ }\left(2,\mathrm{ }-3,\mathrm{ }4\right)$ and $\left(a+1,\mathrm{ }a+2,\mathrm{ }a+3\right)$ respectively. If $AB=5$ and $CD=6$ , then $a=$

The points $A\left(5,-1,1\right);\mathrm{ }B\left(7,-4,\mathrm{ }7\right);\mathrm{ }C\left(1,-6,10\right)$ and $D\left(-1,-3,4\right)$ are vertices of a

The coordinates of point in $XY-$ plane which is equidistant from three points $A\left(2,0,3\right),B\left(0,3,2\right)$ and $C\left(0,0,1\right)$ are

If centroid of tetrahedron $OABC$, where $A, B, C$ are given by $\left(a, 2, 3\right), \left(1, b, 2\right)$ and $\left(2, 1, c\right)$ respectively be $\left(1,\mathrm{ }2,\mathrm{ }-1\right)$ , then distance of $P\left(a, b, c\right)$ from origin is equal to

The points (5, -4, 2), (4, -3, 1), (7, -6, 4) and (8, -7, 5) are the vertices of

The distance of the point $\left(4,3,5\right)$ from the $Y-$ axis is

The points $A\left(5,-1,1\right), B\left(7,-4,7\right), C\left(1,-6,10\right)$ and $D\left(-1,-3,4\right)$ are vertices of a