Discussion
Let \(\alpha\) , \(\beta\) and \(\gamma\) be the angles made by a straight line with the x-axis, y-axis and z-axis respectively. If \(cos\,\alpha + cos\,\beta + cos\,\gamma = \frac{5}{3}\) ,then the value of \(cos\,\alpha\, cos\,\beta + cos\,\beta\, cos\,\gamma + cos\,\gamma\, cos\,\alpha\) is equal to
Answer: B
The value is \(\frac{8}{9}\).
Direction cosines satisfy \(\cos^{2}\alpha + \cos^{2}\beta + \cos^{2}\gamma = 1\). Square the given sum:
\[ \begin{aligned} \left(\tfrac{5}{3}\right)^{2} &= 1 + 2S \\ \tfrac{25}{9} - 1 &= 2S \\ S &= \tfrac{16}{18} = \tfrac{8}{9} \end{aligned} \]
where \(S\) is the required sum of products. Forgetting to halve gives \(\frac{16}{9}\) (not offered); \(\frac{11}{9}\) and \(\frac{7}{9}\) come from using 2 or 3 for the sum of squares.
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