A line makes an angle α, β, γ with the x, y, and z axes. Then sin2 α + sin2 β + sin2 γ is

  1. 1
  2. 0
  3. 2
  4. None of the above

Answer (Detailed Solution Below)

Option 3 : 2
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Detailed Solution

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Concept:

  1. Direction angles: If α, β, and γ are the angles made by the line segment with the coordinate axis then these angles are termed to be the direction angles.
  2. Direction cosines: The cosines of direction angles are the direction cosines of the line. Hence, cos α, cos β and cos γ are called as the direction cosines

It is denoted by l, m and n. ⇔ l = cos α, m = cos β and n = cos γ

  1. The sum of squares of the direction cosines of a line is equal to unity.
  2. l2 + m2 + n2 = 1 or cos2 α + cos2 β + cos2 γ = 1
  3. Direction ratios: Any numbers which are proportional to the direction cosines of a line are called as the direction ratios. It is denoted by ‘a’, ‘b’ and ‘c’.
  4. a ∝ l, b ∝ m and c ∝ n ⇔ a = kl, b = km and c = kn Where k is a constant.

 

Calculation:

We have to find the value of sin2 α + sin2 β + sin2 γ

We know that sum of squares of the direction cosines of a line is equal to unity.

⇒ cos2 α + cos2 β + cos2 γ = 1

⇒ 1 - sin2 α + 1 - sin2 β + 1 - sin2 γ = 1 (∵ sin2 θ + cos2 θ = 1)

⇒ 3 – (sin2 α + sin2 β + sin2 γ) = 1

⇒ 3 – 1 = sin2 α + sin2 β + sin2 γ

∴ sin2 α + sin2 β + sin2 γ = 2

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