Trying to find the right answer to this..

∵ the line makes angles α, β, γ with the axes ∴ cos 2 α + cos 2 β + cos 2 γ = 1 ∴ 1 – sin 2 α + 1 – sin 2 β + 1 – sin 2 γ = 1 or – sin 2 α – sin 2 β – sin 2 γ = –2 ∴ sin 2 α + sin 2 β + sin 2 γ = 2.

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Three Dimensional Geometry

Short Answer Type

11. Find the direction ratios and the direction cosines of the lines joining the pairs of points:
(– 1, –1, –1), (2, 3, 4)

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12. Find the direction cosines of the line passing through the two points (–2, 4, –5) and (1, 2, 3).

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Long Answer Type

13. Find the direction cosines of the sides of the triangle whose vertices are (3, 5, –4), (–1, 1, 2) and (–5, –5, –2).

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Short Answer Type

14.

Find the direction cosines of the line joining the points (2, 1, 2) and (4, 2, 0).

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15.

A line makes angles of 45° and 60° with the positive axes of x and y respectively. What angle does it make with the positive axis of z?

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16. If α, β, γ are the angles which a line makes with the axes, prove that sin²α + sin² β + sin²γ = 2.

∵ the line makes angles α, β, γ with the axes
∴ cos² α + cos² β + cos² γ = 1
∴ 1 – sin² α + 1 – sin²β + 1 – sin² γ = 1 or – sin² α – sin²β – sin² γ = –2
∴ sin² α + sin² β + sin² γ = 2.