The angle between two lines with direction ratios 1, 1, 0 and 1, 0, k is \(\displaystyle\frac{\pi}{3}\). Then the value of k is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\pm 1\)
By the formula for the angle between two lines:
\( \displaystyle \cos\frac{\pi}{3} = \frac{|(1)(1)+(1)(0)+(0)(k)|}{\sqrt{1+1+0}\,\sqrt{1+0+k^2}} \)
\( \displaystyle \frac{1}{2} = \frac{1}{\sqrt{2}\,\sqrt{1+k^2}} \)
\(\sqrt{2(1+k^2)} = 2\)
\(1+k^2 = 2\)
\(k = \pm 1\)
Both signs work, because k appears only as \(k^2\).