Question

A point $P\left( {1,2,3} \right)$ in one vertex of a cuboid formed by the coordinate planes and the planes passing through P and parallel to the coordinate planes. What is the length of one of the diagonals of the cuboid?
A. $\sqrt {10} \,\,{\text{units}}$
B. $\sqrt {14} \,\,{\text{units}}$
C. $4\,\,{\text{units}}$
D. $5\,\,{\text{units}}$

Answer 1

Hint: In this problem, we need to find distance between point P and origin to obtain the length of the diagonal. The formula for the distance between two points in 3D having coordinates (L, M, N) and (A, B, C) is $\sqrt {{{\left( {L - A} \right)}^2} + {{\left( {M - B} \right)}^2} + {{\left( {N - C} \right)}^2}}$.

Complete step by step solution:
Since, the cuboid is formed by the coordinate planes whose parallel planes are passing through point P, the point P and origin will be the two opposite vertices of the cuboid.
The length of the diagonal D is obtained by calculating the distance between point P and origin as shown below.

$$\,\,\,\,\,\,D = \sqrt {{{\left( {1 - 0} \right)}^2} + {{\left( {2 - 0} \right)}^2} + {{\left( {3 - 0} \right)}^2}} \\\ \Rightarrow D = \sqrt {1 + 4 + 9} \\\ \Rightarrow D = \sqrt {14} \\\$$

Thus, the length of the diagonal of the cuboid is $\sqrt {14} \,\,{\text{units}}$, hence, option (B) is the correct answer.

Note: Origin is the opposite vertex of the point P. The formula for the length of the diagonal of a cuboid having length L, breadth B and height H is $\sqrt {{L^2} + {B^2} + {H^2}}$.