Q5. In which octants is the product xyz positive?
The product is positive when no sign or exactly two signs are negative: (+, +, +), (−, −, +), (−, +, −) and (+, −, −), that is octants I, III, VI and VIII.
Q6. The point (a, b, c) lies in octant III. In which octant does (−a, −b, −c) lie?
In octant III, a < 0, b < 0 and c > 0, so (−a, −b, −c) has signs (+, +, −): octant V. Changing all three signs always gives the octant on the far side of O, here V.
Q7. Find the image of (3, −2, 4) in the ZX-plane, and the octants of the point and its image.
The ZX-plane (y = 0) changes the sign of y: the image is (3, 2, 4). The point is in octant IV and the image in octant I.
Q8. Which octants share the half-axis OZ as an edge?
Octants I, II, III and IV, the four above the XY-plane: all of them are bounded by OZ.
Q9. In which octants do the points have x < 0 and z > 0?
II and III, with signs (−, +, +) and (−, −, +).
Q10. On which coordinate plane, or between which two octants, does (0, −4, 7) lie?
x = 0, so it lies on the YZ-plane. With y < 0 and z > 0, it is on the wall between octants III (−, −, +) and IV (+, −, +).
Q11. Write the equations of the plane parallel to the YZ-plane through (6, −1, 2), and of the plane parallel to the XY-plane through the same point.
x = 6 and z = 2. A plane parallel to a coordinate plane fixes one coordinate, the one that is 0 on that coordinate plane.
Q12. A point is on the XY-plane and on the YZ-plane. What can you say about it?
It has z = 0 and x = 0, so it is on the y-axis: it has the form (0, y, 0).