Introduction to Three Dimensional Geometry

The position of a bird in the sky needs three numbers, not two. This chapter extends coordinate geometry from the plane to space. Three mutually perpendicular lines through a point O, called the x-axis, y-axis and z-axis, form a rectangular coordinate system. These axes determine three coordinate planes, the XY-plane, YZ-plane and ZX-plane, which divide space into eight parts called octants. Every point P in space is described by an ordered triple (x, y, z), its perpendicular distances from the YZ, ZX and XY planes. You will learn how the signs of the coordinates tell us the octant in which a point lies. For example, (2, -3, 4) is in the octant XOY'Z. A point on the x-axis has the form (x, 0, 0). The chapter then derives the distance formula between two points P(x1, y1, z1) and Q(x2, y2, z2): PQ = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2). You will use it to test collinearity, to classify triangles and parallelograms, and to find simple loci.

Detailed Worksheet: Introduction to Three Dimensional Geometry Section A - Definitions (10 marks) 1. Name the three coordinate planes and state the equation of each. (2 marks) 2. How many octants does a three-dimensional coordinate system have? Name the octant in which (-3, 1, 2) lies. (2 marks) 3. What are the coordinates of a point on the y-axis? Give an example. (2 marks) 4. Find the distance of the point (2, -3, 6) from the origin. (2 marks) 5. Find the distance between the points (2, 3, 5) and (4, 3, 1). (2 marks) Section B - Calculations and Applications (15 marks) 6. Show that the points (-2, 3, 5), (1, 2, 3) and (7, 0, -1) are collinear. (3 marks) 7. Verify that (0, 7, -10), (1, 6, -6) and (4, 9, -6) are the vertices of an isosceles right-angled triangle. (3 marks) 8. Find the point on the x-axis which is equidistant from (1, 2, 3) and (3, 5, -2). (3 marks) 9. Find the equation of the set of points P which are equidistant from (1, 2, 3) and (3, 2, -1). (3 marks) 10. Find the points on the y-axis which are at a distance of 13 units from (3, 0, 4). (3 marks) Section C - Diagrams (10 marks) 11. Draw the three coordinate axes and mark the points (2, 1, 3), (-1, 2, 1) and (3, -2, -1) in space. (4 marks) 12. Draw a figure showing the eight octants formed by the coordinate planes and label them with the signs of the coordinates. (3 marks) 13. Draw a cuboid with one vertex at the origin and edges along the axes, of lengths 3, 4 and 5. Write the coordinates of all its vertices. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Verify that (-1, 2, 1), (1, -2, 5), (4, -7, 8) and (2, -3, 4) are the vertices of a parallelogram. (5 marks) 15. Find the equation of the set of points P such that the sum of the distances of P from A(4, 0, 0) and B(-4, 0, 0) is 10. (5 marks) 16. Show that the points A(1, 2, 3), B(-1, -2, -1), C(2, 3, 2) and D(4, 7, 6) are the vertices of a parallelogram ABCD, but not a rectangle. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Draw neat figures with labelled axes. Show all steps and simplify surds.