Question

If projections of any line on coordinate axes are 3,4 and 5, then what is its length?
A. $12$
B. $50$
C. $5\sqrt 2 $
D. $3\sqrt 2 $

Answer 1

Hint: For a line of any length, say $d$, its projection on any axis is calculated by taking the product of its length and the cosine of the angle, the line formed with the axis itself. Now, these cosines form the direction cosines of the line. Thus, the projection of a line on any axis is $d$ multiplied by the direction cosine along that axis respectively.

Complete step by step Solution:
Let the length of the line be $d$ and let the direction cosines be $(l,m,n)$ respectively.
As direction cosines of a coplanar vector, in space, are the cosines of the angles made by the line with the coordinate axes, therefore,
$dl$ is the projection of the line on the x-axis,
$dm$ is the projection of the line on the y-axis, and
$dn$ is the projection of the line on the z-axis.
The projections of the line on coordinate axes are 3,4 and 5 respectively.
Hence,
$dl = 3$ … (1)
$dm = 4$ … (2)
$dn = 5$ … (3)
Squaring and adding equations (1), (2), and (3) together,
${d^2}{l^2} + {d^2}{m^2} + {d^2}{n^2} = 9 + 16 + 25$
Simplifying further,
${d^2}({l^2} + {m^2} + {n^2}) = 50$
As the sum of the squares of the direction cosines of a line equals to 1, that is, ${l^2} + {m^2} + {n^2} = 1$ ,
Therefore, substituting this value,
${d^2} = 50$
Taking the square root,
$d = \pm 5\sqrt 2 $
As length cannot be negative, hence, $d = 5\sqrt 2 $

Hence, the correct option is (C).

Note: Direction cosines of any coplanar vector, in space, determine the direction of the vector. They are the cosines of the angle, the line made with the x-axis, y-axis, and z-axis. Let $(l,m,n)$ be the direction cosines of any line, then, ${l^2} + {m^2} + {n^2} = 1$ .