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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).

191 Views

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).

145 Views

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.

Let P(– 1, 2, – 3), Q(4, 5, 1) , R(9, 8, 5) be given points.
The direction ratios of PQ are 4 + 1, 5 – 2, 1 + 3 i.e. 5, 3, 4,
The direction ratios of PR are 9 + 1, 8 – 2, 5 + 3, i.e. 10, 6, 8.
Since $10 over 5 space equals 6 over 3 space equals space 8 over 4$
∴ lines PQ and PR are parallel.
But P is a common point on both lines.
∴ points P, Q, R 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.

93 Views