Presentation on theme: "The Three-Dimensional Coordinate System 11.1"— Presentation transcript:
1 The Three-Dimensional Coordinate System 11.1 JMerrill, 2010
2 Solid Analytic Geometry The Cartesian plane (rectangular coordinate system) is determined by 2 perpendicular number line (x- and y-axis) and their point of intersection (the origin).To identify a point in space, we need a third dimension. The geometry of this three-dimensional model is called solid analytic geometry.The 3-D coordinate system is formed by passing a z-axis perpendicular to both the x- and y-axes at the origin.
3 Coordinate PlanesNotice we draw the x- and y-axes in the opposite directionX = directed distance from yz-plane to some point PY= directed distance from xz-plane to some point PZ= directed distance from xy-plane to some point P(x,y,z)So, to plot points you go out, over, up/down
4 OctantsThe 3-D system can have either a right-handed or a left-handed orientation.We’re only using the right-handed orientation meaning that the octants (quadrants) are numbered by rotating counterclockwise around the positive z-axis.There are 8 octants.
6 Plotting Points in Space Plot the points:(2,-3,3)(-2,6,2)(1,4,0)(2,2,-3)Draw a sideways x, then put a perpendicular line through the origin.
7 FormulasYou can use many of the same formulas that you already know because right triangles are still formed.
8 The Distance FormulaIt looks the same in space as it did before except with a third coordinate:
9 ExampleFind the distance between (1, 0, 2) and (2, 4,-3)
10 Midpoint Formula The midpoint formula is What is the midpoint if you make a 100 on a test and an 80 on a test?So the midpoint is just the average of the x’s, y’s, and z’s.
11 Midpoint You DoFind the midpoint of the line segment joining (5, -2, 3) and (0, 4, 4)
12 Equation of a Sphere The equation of a circle is x2 + y2 = r2 If the center is not at the origin, then the equation is (x-h)2 + (y-k)2 = r2The equation of a sphere whose center is at (h,k,j) with radius r is (x-h)2 + (y-k)2 + (z–j)2= r2
13 Finding the Equation of a Sphere Find the standard equation of a sphere with center (2,4,3) and radius 3(x-h)2 + (y-k)2 + (z–j)2= r2(x-2)2 + (y-4)2 + (z–3)2 = 32Does the sphere intersect the plane?Yes. The center of the sphere is 3 units above the y-axis and has a radius of 3. It intersects at (2,4,0).
14 Finding the Center and Radius of a Sphere Find the center and radius of the sphere given by x2 + y2 + z2 – 2x + 4y – 6z +8 = 0This works the same way as it did in 2-D space. In order to find the center, we must put the equation into standard form, which means completing the square.
15 Finding the Center and Radius of a Sphere x2 + y2 + z2 – 2x + 4y – 6z +8 = 0(x-1)2 + (y+2)2 + (z-3)2 = 6The center is (1,-2,3) and the radius is √6.