Question

If the direction cosines of a variable line in two adjacent positions be l, m, n and l + a, m + b, n + c and the small angle between the two positions be $\theta$, then :

• A. $\theta =a+b+c$
• B. ${\theta }^2=a^2+b^2+c^2$
• C. $|\theta |=|a|+|b|+|c|$
• D. ${\theta }^3=a^3+b^3+c^3$

Answer 1

The correct option is B ${\theta }^2=a^2+b^2+c^2$

$l^2+m^2+n^2=1,(l+a{)}^2+(m+b{)}^2+(n+c{)}^2=1\Rightarrow al+bm+cn=-\frac{1}{2}(a^2+b^2+c^2)$

$cos\theta =(a+l)l+m(b+m)+n(c+n)$
= 1 + al + bm + cn
$\Rightarrow a^2+b^2+c^2=2(1-cos\theta )=4si{n}^2\left(\frac{\theta }{2}\right)$
Since $\theta$ is small, $sin\left(\frac{\theta }{2}\right)\approx \frac{\theta }{2}$
$\therefore {\theta }^2=a^2+b^2+c^2[\because si{n}^2(\frac{\theta }{2})\approx \frac{{\theta }^2}{4}]$