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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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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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14.

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

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Let a, β, γ be the angles which line makes with axes.

therefore

direction - cosines of the line are

cos space 45 degree comma space space cos space 60 degree comma space space cos space straight gamma

open square brackets because space space space straight alpha space equals space 45 degree comma space space straight beta space equals space 60 degree close square brackets

therefore

cos squared 45 degree plus cos squared 60 degree plus space cos squared straight gamma space equals space 1 space space space space space space space space space space space open square brackets because space space cos squared straight alpha plus cos squared straight beta plus cos squared straight gamma space equals space 1 close square brackets

rightwards double arrow space space space space space 1 half plus 1 fourth plus cos squared straight gamma space equals space 1 space space space space rightwards double arrow space space space cos squared straight gamma space equals space 1 fourth space space rightwards double arrow space space cos space straight gamma space equals space plus-or-minus 1 half space rightwards double arrow space straight gamma space equals space 60 degree space space or space space 120 degree

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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.

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17.

Show that the points (2, 3, 4), (–1, –2, 1), (5, 8, 7) are collinear.

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18. Show that the points (–1, 2, –3), (4, 5, 1) and (9, 8, 5) are collinear.

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19. Show that the points A(2, 3, – 4), B(1, –2, 3) and C (3, 8, –11) are collinear.

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20.

Find the length of the projection of the line segment joining the points P (3, -1, 2) and Q(2, 4, -1) on the line with direction ratios -1, 2, -2.

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