Question

If the vectors $\overrightarrow{b}$ (tan a, – 1, 2 $\sqrt{\sin\alpha/2}$) and
$\overrightarrow{c}$ = (tan a, tan a, – $\frac{3}{\sqrt{\sin\alpha/2}}$) are orthogonal and a vector $\overrightarrow{a}$ = (1, 3, sin 2a) makes an obtuse angle with z-axis, then the value of a is –

• A. a = (4n + 1) p – tan^–1 2
• B. a = (4n + 2) p – tan^–1 2
• C. a = (4n + 1) p + tan^–1 2
• D. a = (4n + 2) p + tan^–1 2

Answer 1

Answer: (A)

a = (4n + 1) p – tan^–1 2

Answer 2

$\overrightarrow{a}$ = (1, 3, sin 2a) makes an obtuse angle with z-axis

sin 2 a < 0

$\overrightarrow{b}$ and $\overrightarrow{c}$ are orthogonal $\overrightarrow{b}$.$\overrightarrow{c}$ = 0

̃ tan² a – tan a – 6 = 0

̃ tan a = 3 or – 2

If tan a = 3

sin 2a = $\frac{2\tan\alpha}{1 + \tan^{2}\alpha}$ = $\frac{3}{5}$ > 0 (not possible),

tan a = – 2

tan 2 a = $\frac{4}{3}$ > 0

sin 2a < 0

2a lies in the third quadrant ̃ $\frac{\alpha}{2}$ lies in 2^nd quadrant \ $\sqrt{\sin\alpha/2}$ is valid and

a = (4n + 1) p – tan^–1 2.