A line  AB in three-dimensional space makes angles 450 and 120 with the positive x-axis and the positive y-axis respectively. If AB makes an acute angle \Theta with the positive z-axis , than \Theta equals

  • Option 1)

    300

  • Option 2)

    450

  • Option 3)

    600

  • Option 4)

    750

 

Answers (1)

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)

-

 

 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

Incorrect answer

Option 2)

450

Incorrect answer

Option 3)

600

Correct answer

Option 4)

750

Incorrect answer

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