Let PM be the perpendicular from the point P(1, 2, 3) to x-y plane. If OP makes an angle θ with the positive direction of the z-axis and OM makes an angle φ with the positive direction of x- axis, where O is the origin, then find θ and φ .
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
Sol. Here, P be (x, y, z) shown as;
then, x = r sin θ .cos φ , y = r sin θ sin φ , z = r cos θ
⇒ 1 = r sin θ . cos φ , 2 = r sin θ sin φ , 3 = r cos θ … (i)
⇒ 1 2 + 2 2 + 3 2 = r 2 sin 2 θ cos 2 φ + r 2 sin 2 θ sin 2 φ + r 2 cos 2 θ
= r 2 sin 2 θ (cos 2 φ + sin 2 φ ) + r 2 cos 2 θ
= r 2 sin 2 θ + r 2 cos 2 θ = r 2 ⇒ r = ± 
∴ from (i), we have
sin θ cos φ = + 
, sin θ sin φ = 
, cos θ = 
(neglecting –ve sign as acute angles)
∴ 
= 
and tan θ = 
= 
⇒ tan φ = 2 and tan θ = 
/3
⇒ φ = tan
–1 2 and θ = tan –1 ( 
/3)
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