Under what condition do ( 1 2 , 1 2 , k ) represent direction cosines of a line?

Question

Under what condition do $(\frac{1}{\sqrt{2}}, \frac{1}{2}, k)$ represent direction cosines of a line?

Smart Achievers · Accepted Answer

(2)

Ans. $k = \pm \frac{1}{2}$

Explanation:

For $(\frac{1}{\sqrt{2}}, \frac{1}{2}, k)$ to represent direction cosines, we should have

${(\frac{1}{\sqrt{2}})}^{2} + {(\frac{1}{2})}^{2} + k^{2} = 1$

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

$k^{2} = 1 - \frac{3}{4}$

$\Rightarrow$ $k = \pm \frac{1}{2}$

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