Study Sheet (13.1-13.2)
The Three-Dimensional Coordinate System
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 threedimensional model is called solid analytic geometry.
•
The 3-D coordinate system is formed by passing a z-axis perpendicular to both the x- and yaxes at the origin.
Coordinate Planes
Notice we draw the x- and y-axes in the opposite direction
X = directed distance from yz-plane to some point P
Y= directed distance from xz-plane to some point P
Z= directed distance from xy-plane to some point P
So, to plot points you go out, over, up/down
Octants
•
The 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.
Plotting Points in Space
Plot the points: (2,-3,3), (-2,6,2), (1,4,0), and (2,2,-3)
Draw a sideways x, then put a perpendicular line through the origin.
Formulas
You can use many of the same formulas that you already know because right triangles are still
formed.
The Distance Formula
•
It looks the same in space as it did before except with a third coordinate:
Example1: Find the distance between (1, 0, 2) and (2, 4,-3).
Midpoint Formula
The midpoint formula
is
So the midpoint is just the average of the x’s, y’s, and z’s.
Example 2: Find the midpoint of the line segment joining (5, -2, 3) and (0, 4, 4).
Equation of a Sphere
Recall: 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 = r2
The equation of a sphere whose center is at (h, k , l ) with radius r is
Finding the Equation of a Sphere
Example 3: Find the standard equation of a sphere with center (2,4,3) and radius 3.
Does the sphere intersect the plane?
Finding the Center and Radius of a Sphere
Example 4: Find the center and radius of the sphere given by x2 + y2 + z2 – 2x + 4y – 6z +8 = 0
Example 5: Finding the Center and Radius of a Sphere
(a) x2 + y2 + z2 – 2x + 4y – 6z +8 = 0
(b) (x-1)2 + (y+2)2 + (z-3)2 = 6
(c) The center is (1,-2,3) and the radius is √6.
Study Sheet (13.4)
Functions of Several Variables
Understanding Functions of Two Variables
Consider the function C ( x, y) = x 2 − 3 y .
• Use a table to investigate this function for x = 0, 1, 2, 3 and y = 0, 1, 2, 3.
• Use your calculator to sketch this function assuming y is a constant.
• Use your calculator to sketch this function assuming x is a constant.
Partial Derivatives
Computing Partial Derivatives Algebraically
Example:
Find:
(a) f x and
(b)
f y if f (x, y) = 2x 2 + 3y 2 .
∂P
if
∂r
(c) z x if
P = 100ert
z = x 2 y + 2x 5 y
Example: Find f xy and
f yx if f (x, y) = x 2 + xy + 3y .