JEE Main · 2010 · 11 Apr 2010 · Offline · Three Dimensional Geometry

A line AB in three-dimensional space makes angles

4 5^{\circ}

and

12 0^{\circ}

with the positive x-axis and the positive y-axis respectively. If AB makes an acute angle

θ

with the positive z-axis, then

θ

equals

(A) $4 5^{\circ}$
(B) $6 0^{\circ}$ ✓ Correct
(C) $7 5^{\circ}$
(D) $3 0^{\circ}$

Answer: (B)

Solution

The direction cosines of a line satisfy

ℓ^{2} + m^{2} + n^{2} = 1

. Here

ℓ = cos 4 5^{\circ} = \frac{1}{2}

and

m = cos 12 0^{\circ} = - \frac{1}{2}

, so

\frac{1}{2} + \frac{1}{4} + cos^{2} θ = 1

, giving

cos^{2} θ = \frac{1}{4}

. As

θ

is acute,

cos θ = \frac{1}{2}

and

θ = 6 0^{\circ}

JEE Main · 2010 · 11 Apr 2010 · Offline · Single correct option

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