# A line  AB in three-dimensional space makes angles 450 and 1200  with the positive $x$-axis and the positive $\dpi{100} y$-axis respectively. If AB makes an acute angle $\dpi{100} \Theta$ with the positive $\dpi{100} z$-axis , than $\dpi{100} \Theta$ equals Option 1) 300 Option 2) 450 Option 3) 600 Option 4) 750

P Plabita

As we learnt in

Direction Cosines -

i)    $\sin^{2} \alpha+ \sin^{2} \beta+\sin^{2} \gamma= 2$

ii)    If OP =r then the co-ordinates of P will be (lr,mr,nr)

iii)    Direction cosines of X-axis are (1,0,0)

iv)    Direction cosines of Y-axis are (0,1,0)

v)    Direction cosines of Z-axis are (0,0,1)

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Direction angles,

Given $\alpha= 45^{^{0}}, \beta= 120^{0}$

We know that,

$\cos ^{2}\alpha + \cos ^{2}\beta +\cos ^{2}\gamma = 1$

$\frac{1}{2}+\frac{1}{4} + \cos ^{2}\gamma =1$

$\cos ^{2}\gamma =\frac{3}{4}$

$\cos \gamma =\frac{\sqrt{3}}{2}$

Option 1)

300

Option 2)

450

Option 3)

600