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Section 11.1

Section 11.1. 3 Dimensional Coordinate Plane. We can expand our 2-dimensional ( x-y ) coordinate system into a 3-dimensional coordinate system , using x- , y- , and z -axes.

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Section 11.1

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  1. Section 11.1 3 Dimensional Coordinate Plane

  2. We can expand our 2-dimensional (x-y) coordinate system into a 3-dimensional coordinate system, using x-, y-, and z-axes.

  3. The x-y plane is horizontal in our diagram and shaded green. It can also be described using the equation z = 0, since all points on that plane will have 0 for their z-value. • The x-z plane is vertical and shaded pink. This plane can be described using the equation y = 0. • The y-z plane is also vertical and shaded blue. The y-z plane can be described using the equation x = 0.

  4. yz plane x = 0 xz plane y = 0 xy plane z = 0

  5. These three coordinate planes separate the three-dimensional coordinate system into eight octants. z y x

  6. We normally use the 'right-hand orientation' for the 3 axes, with the positive x-axis pointing in the direction of the first finger of our right hand, the positive y-axis pointing in the direction of our second finger and the positive z-axis pointing up in the direction of our thumb.

  7. In a right-handed system, Octant I is the octant where all coordinates are positive. Octant II, III, IV are found by rotating counterclockwise around the positive z-axis. • Octant V is below Octant I and has coordinates of (x, y, -z). Octants VI, VII, and VIII are then found by rotating counterclockwise around the negative • z-axis.

  8. III z VII IV II I y VI x VIII V

  9. Example 1 • In 3-dimensional space, graph the point • (2, 3, 5). • To reach the point (2, 3, 5), we move 2 units along the x-axis, then 3 units in the y-direction, and then up 5 units in the z-direction.

  10. Distance in 3-dimensional Space • To find the distance from one point to another in 3-dimensional space, we just extend the Pythagorean Theorem.

  11. Distance Between 2 Points in 3 Dimensions • If we have point A (x1, y1, z1) and another point B (x2, y2, z2) then the distance AB between them is given by the formula:

  12. Example 2 • Find the distance between the points • P (2, 3, 5) and Q (4, -2, 3).

  13. Midpoint Formula in 3 Dimensions • The midpoint of the line segment joining the points (x1, y1, z1) and (x2, y2, z2) is

  14. Example 3 • Find the midpoint of the line segment joining (2, -2, 4) and (1, 3, 6).

  15. Sphere • A sphere with center (h, k, j) and radius r is defined as the set of all points (x, y, z) such that the distance between (x, y, z) and • (h, k, j) is r.

  16. Standard Equation of a Sphere • The standard equation of a sphere with center (h, k, j) and radius r is given by

  17. Example 4 • Find the standard equation of the sphere with center (1, 5, -2) and radius 4. Does this sphere intersect the xy-plane? • (x – 1)2 + (y – 5)2 + (z + 2)2 = 16 • The sphere does intersect the xy-plane since it is only 2 units below this plane and the radius of the sphere is 4.

  18. Example 5 • Find the center and radius of the sphere given by • x2 + y2 + z2 +4x – 2y + 8z + 10 = 0 • (x + 2)2 + (y – 1)2 + (z + 4)2 = 11 • center: (-2, 1, -4)

  19. Finding the intersection of a surface with one of the three coordinate planes (or with a plane parallel to one of the three coordinate planes) helps us visualize the surface. • Such an intersection is called a trace of the surface • The xy-trace of a surface consists of all points that are common in both surface and the xy-plane.

  20. Example 6 • a. What shape is the xy-trace of the sphere given by • (x – 2)2 + (y – 3)2 + (z + 6)2 = 49 • It is a circle. • b. Find the equation of the xy-trace. • (x – 2)2 + (y – 3)2 = 13

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