S3 Mathematics Chapter 3: 3D Figures Projection of Points and Lines Onto Plane Chapter Objective 2 Projection of points and lines onto planes Angles in 3D Chapter Objective 2 Projection of points and lines onto planes Projection onto Planes Textbook Page 3.14 A’ is the projection of point A onto the plane, while B is not. 4 Projection onto Planes Textbook Page 3.14 A’ is the projection of point A onto the plane, because AA’ is perpendicular to the plane. B is not the projection of A onto the plane, because AB is not perpendicular to the plane. 5 Projection onto Planes Textbook Page 3.14 A line can also be projected on a plane in a similar way. 6 Angle between Line and Plane Textbook Page 3.16 Angle between line L on plane P = angle between L and its projection on P 7 Two Intersecting Planes Textbook Page 3.16 Two distinct non-parallel planes will intersect at a line. 8 Angle between two planes 1. 2. 3. 4. Textbook Page 3.16 Find the line of intersection L. Choose a point (A) on the intersection line. On each plane, draw lines L1, L2 from A and perpendicular to L. The angle between L1, L2 is the angle between two planes. 9 Angle between two planes 1. 2. 3. 4. Textbook Page 3.16 Find the line of intersection L. Choose a point on the intersection line (A). On each plane, draw lines L1, L2 from A and perpendicular to L. The angle between L1, L2 is the angle between two planes. 10 Example: Projection of line The figure shows a cuboid. What is the projection of EB onto plane ABCD? A AE B DB C EC D CD Example: Projection of line The figure shows a cuboid. What is the projection of EB onto plane ADEF? A AE B DB C EC D CD Example 1: Angle between planes The diagram shows a right pyramid with a square base ABCD. P, Q and R are mid points of AB, AD and DC respectively. M is the projection of V onto base ABCD. Which of the following is the angle between plane VAB and its base ABCD? A ∠ VBC B ∠ VBD C ∠ VPM D ∠ VND Example 1: Angle between planes The diagram shows a right pyramid with a square base ABCD. P, Q and R are mid points of AB, AD and DC respectively. M is the projection of V onto base ABCD. Which of the following is the angle between plane VAB and its base ABCD? A ∠ VBC B ∠ VBD C ∠ VPM D ∠ VND Example 1: Angle between planes What is the angle between planes VAB and VAD? Example 1: Angle between planes What is the angle between planes VAB and VAD? Example 2: Angle between planes The diagram shows a pyramid with a rectangular base ABCD. N is the mid point of BC. VD is the height of the pyramid. Which of the following is the angle between plane VBC and its base ABCD? A ∠ VBA B ∠ VBD C ∠ VCD D ∠ VND Example: Angle between planes Answer: ∠ VCD Note that VD is the height of the pyramid. Example: Angle between planes Name the angle between plane VAD and VDC? Example: Angle between planes Name the angle between plane VAD and VDC? ∠ ADC Example: Angle between planes Name the angle between plane VAB and VDC? Example: Angle between planes Name the angle between plane VAB and VDC? ∠ AVD Example: Angle between planes Name the angle between plane VAB and VBC? Example: Angle between planes Name the angle between plane VAB and VBC? ∠ AXQ or CYR
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