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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).

The direction ratios of the line joining the points (–2, 4, –5) and (1. 2, 3) are
1 + 2, 2 – 4, 3 + 5 i.e. 3, – 2, 8.
Dividing each by $square root of left parenthesis 3 right parenthesis squared plus left parenthesis negative 2 right parenthesis squared plus left parenthesis 8 right parenthesis squared end root space equals space square root of 9 plus 4 plus 64 end root space equals space square root of 77 comma$ we get the direction cosines of the line as $fraction numerator 3 over denominator square root of 77 end fraction comma space minus fraction numerator 2 over denominator square root of 77 end fraction comma space fraction numerator 8 over denominator square root of 77 end fraction.$

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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.

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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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