Question

If the direction cosines of a line are $-\frac{1}{2},\frac{1}{3}$ k then the value of k is

• A. $\frac{3}{2}$
• B. $\frac{2}{3}$
• C. $\frac{\sqrt{23}}{6}$
• D. $\frac{23}{3}$

Answer 1

The correct option is C $\frac{\sqrt{23}}{6}$

The sum of the squares of direction cosines is one.

$i.e.$ ${\cos }^2\alpha +co{s}^2\beta +{\cos }^2\gamma =1$

Given that the direction cosines of a line are $-\frac{1}{2},\frac{1}{3},k$

$\Rightarrow$ ${(-\frac{1}{2})}^2+{\left(\frac{1}{3}\right)}^2+k^2=1$

$\Rightarrow$ $\frac{1}{4}+\frac{1}{9}+k^2=1$

$\Rightarrow$ $k^2=1-\frac{1}{4}-\frac{1}{9}$

$\Rightarrow$ $k^2=\frac{36-9-4}{36}$

$\Rightarrow$ $k^2=\frac{23}{36}$

$\Rightarrow$ $k=\frac{\sqrt{23}}{6}$