Question

If $l,m,n$ are the direction cosines of a vector if $l=\frac{1}{2}$ ,then the maximum value of $lmn$ is

• A. $\frac{1}{4}$
• B. $\frac{3}{8}$
• C. $\frac{1}{2}$
• D. $\frac{3}{16}$

Answer 1

Answer: (D)

$\frac{3}{16}$

$l,m,n$ are direction cosines of a vector so $l^2+m^2+n^2=1$
$m^2+n^2=\frac{3}{4}$
$(m-n{)}^2+2mn=\frac{3}{4}$
$mn=\frac{3}{8}\frac{-1}{2}(m-n{)}^2$

$lmn=\frac{3}{16}\frac{-1}{4}(m-n{)}^2$

$lmn\le \frac{3}{16}$
So max $lmn=\frac{3}{16}$