Problems : Three Dimensional Geometry

Level – I

1. Find the direction cosines of the line which is perpendicular to the lines with direction cosines proportional to (1, – 2, – 2), (0, 2, 1)

2. Find the area of the triangle whose vertices are (0, 0, 0), (3, 4, 7) and (5, 2, 6)

3. Find the point common to the lines which join (6, – 7, 0) to (16, – 19, – 4) and (0, 3, – 6) to (2, – 5, 10).

4. Find the lengths of the edges of the rectangular parallelepiped formed by planes drawn through (1, 2, 3) and (4, 7, 6) parallel to coordinate planes.

5. A (6, 3, 2) B (5, 1, 4) C (3, – 4, 7) D (0, 2, 5) are four points. Find the projection of the segment AB on the line CD and segment CD on the line AB.

6. The plane ax + by + cz = 1 meets the axes OX, OY, OZ in A, B, C. A plane through the x-axis bisects angle A of the triangle ABC. Similarly planes through the other two axes bisect the angles B and C. Find the equations of the line of intersection of these planes.

7. Find the equation of the line passing through (1, 1, 1) and perpendicular to the line of intersection of the planes x + 2y – 4z = 0 and 2x – y + 2z = 0.

8. Find the equation of a plane passing thorough the line

and making an angle of 300 with the plane x + y + z = 5.

9. Find the equation of the plane which bisects the join of P(x1 , y1 , z1) and Q(x2 , y2 , z2) perpendicularly.

10. Prove that the three lines from O with direction cosines l1 , m1 , n1 ; l2 , m2 , n2 ; l3 , m3 , n3 are coplanar if

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