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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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The direction ratios of the line joining (2, 1, 2) and (4, 2, 0) are 4 – 2, 2 – 1, 0 – 2 i.e., 2, 1, –2 i.e.,
Dividing each by

square root of left parenthesis 2 right parenthesis squared plus left parenthesis 1 right parenthesis squared plus left parenthesis negative 2 right parenthesis squared end root space equals space 3 comma

the direction-cosines of the line are

2 over 3 comma space 1 third comma space minus 2 over 3.

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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.

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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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