Question

If l, m, n are the direction cosines of vector if \[l=\dfrac{1}{2}\], then the maximum value of lmn is
(a)\[\dfrac{1}{4}\]
(b)\[\dfrac{3}{8}\]
(c)\[\dfrac{1}{2}\]
(d)\[\dfrac{3}{16}\]

Answer 1

Hint: The direction cosine are given by the relation \[{{l}^{2}}+{{m}^{2}}+{{n}^{2}}=1\], put value of l in it. For terms m and n the arithmetic mean & geometric mean are of relation A.M \[\ge \] G.M, thus get max value for mn according to the relation and find max value of lmn.

Complete step-by-step answer:
It is said that l, m and n are the direction cosines of a vector.
We have been given that, \[l=\dfrac{1}{2}\].
We need to find the maximum value of lmn.
The direction cosines of a line parallel to any coordinate axis are equal to the direction cosine of the corresponding axis.
The direction cosines are associated by the relation,
\[\Rightarrow {{l}^{2}}+{{m}^{2}}+{{n}^{2}}=1\]
Now let us put \[l=\dfrac{1}{2}\] in the above expression. Thus we get,
\[\Rightarrow {{\left( \dfrac{1}{2} \right)}^{2}}+{{m}^{2}}+{{n}^{2}}=1\]
\[\begin{align}
  & \Rightarrow {{m}^{2}}+{{n}^{2}}=1-\dfrac{1}{4}=\dfrac{4-1}{4} \\
 & \therefore {{m}^{2}}+{{n}^{2}}=\dfrac{3}{4} \\
\end{align}\]
We are considering Arithmetic mean and geometric mean.
The arithmetic mean is always greater than geometric mean i.e. A.M \[\ge \] G.M.
We know that for two numbers m and n,
Arithmetic mean, A.M = \[\dfrac{{{m}^{2}}+{{n}^{2}}}{2}\].
Geometric mean, G.M = \[\sqrt{{{m}^{2}}+{{n}^{2}}}\] = mn.
\[\therefore \] A.M \[\ge \] G.M
\[\Rightarrow \dfrac{{{m}^{2}}+{{n}^{2}}}{2}\ge \sqrt{{{m}^{2}}+{{n}^{2}}}\]
Put, \[{{m}^{2}}+{{n}^{2}}\] = \[\dfrac{3}{4}\].
i.e. We get \[\dfrac{3}{4\times 2}\ge \sqrt{{{m}^{2}}{{n}^{2}}}\]
Hence we got \[\dfrac{3}{8}\ge mn\].
Hence we got the maximum value of mn as \[\dfrac{3}{8}\]. What we need is the maximum value of lmn.
\[\therefore \] Maximum value of lmn = l \[\times \] maximum value of mn
Maximum value of lmn = \[\dfrac{1}{2}\times \dfrac{3}{8}=\dfrac{3}{16}\].
Hence we got the maximum value of lmn = \[\dfrac{3}{16}\].
\[\therefore \] Option (d) is the correct answer.

Note: The direction cosines of the x, y and z axis are given as (1, 0, 0), (0, 1, 0) and (0, 0, 1). In case of parallel lines, the direction cosines are always the same. A line can have two sets of direction cosine according to the direction.