Download 7.3 - 3D Template

7.3 – 3-D Notes Honors Precalculus
Adapted from 11.1 & 11.4
Date: __________________________________
The Three-Variable Coordinate System
I. Cartesian Plane
The familiar xy-coordinate system is used to represent pairs of numbers (ordered pairs (x, y) ) by points in the plane.
For example, show how we might represent the combination of values x = 2 and y = -3.
II. Three-Dimensional Coordinate System
A similar system can be used to represent triples of numbers (ordered triples - (x, y, z) ).
One common way to do this is to picture the x-axis and y-axis as “lying flat” and the zaxis as “going up” perpendicular to that plane.
In this section, we consider the positive direction of the z-axis to be straight up, the
positive direction of the x-axis as coming out of the page, and the positive direction of
the y-axis as going to the right.
As in the two-variable system, the point where the three axes meet is called the ________________ and represents the
values x = __________, y = __________, and z = __________. We write this point simply as (_____, _____, _____).
The collection of all points in this system is called the three-dimensional coordinate system and is often referred to simply
as 3-space. Each pair of axes defines a plane, and these are known as the coordinate planes. In the diagram below, the
first plane is called the xy-plane, the second plane is called the xz-plane, and the third plane is called the yz-plane.
These planes divide 3-space into eight separate regions known as octants. (The octants are analogous to the quadrants of
the two-variable coordinate system.) Although there is no standard numbering for all of the octants, the set of points whose
coordinates are all positive is called the first octant.
The point (2, -3, 4) is found as shown in the diagram at the right, by going two
units “toward you” from the origin, 3 units to the left and 4 units up. Try
graphing this on the isometric dot paper.
III. Graphing a Plane in 3D
Trace
To sketch a plane in space, it is helpful to find its points of intersection with the coordinate axes and its ________________
in the coordinate planes. Find the ____________________ of each axes.
Ex. 1
Find the trace on each of the coordinate planes. Find the intercepts of each axes. Using isometric dot paper, graph
the plane 3x + 2y + 4z = 12 .
If the equation of a plane has a missing variable, the plane must be ____________________ to the axis represented by the
___________________ variable.
Ex. 2
Determine which coordinate axis 2x + z = 1 is parallel to.
If __________ variables are missing from the equation of a plane, then it is ____________________ to the
___________________ plane represented by the missing variables.
Ex. 3
Determine which coordinate plane 5y = 1 is parallel to.
7.3 Notes Honors Precalculus
Date: __________________________________
I. Multivariable Linear Systems: Row-Echelon Form & Back Substitution
System of Three Linear
Equations in Three Variables:
⎧ x − 2y + 3z = 9
Row-Echelon Form
⎪
= −4
⎨ −x + 3y
⎪ 2x − 5y + 5z = 17
⎩
Equivalent System in RowEchelon Form:
⎧ x − 2y + 3z = 9
⎪
y + 3z = 5
⎨
⎪
z=2
⎩
Ordered Triple
Ex. 1
⎧ 2x − y + 5z = 22
⎪
Use back-substitution in row-echelon form to solve this system of linear equations ⎨
y + 3z = 6 .
⎪
z=3
⎩
II. Multivariable Linear Systems: Gaussian Elimination
Two systems are ________________________ if they have the __________ solution set.
To solve a system that is not in row-echelon form, first convert it to an ____________________ system that is in rowechelon form by using the following operations:
Operations That Produce Equivalent Systems
Each of the following row operations on a system of linear equations produces an equivalent system of linear equations.
1. ______________________ two equations.
2. ______________________ one of the equations by a nonzero constant.
3. ______________________ a multiple of one of the equations to another equation to replace the latter equation.
For a system of linear equations, exactly one of the following is true:
Number of
Solutions of a
Linear System
1. There is exactly __________ solution.
2. There are _______________ _______________.
3. There is __________ solution.
Ex. 2
⎧⎪ 2x + y = 90
Solve the system of equations ⎨
.
⎪⎩ x + 2y = 90
Ex. 3
⎧ x−y+z=4
⎪
Solve the system of equations ⎨ x + 3y − 2z = −3
⎪ 3x + 2y + 2z = 6
⎩
Ex. 4
⎧ x − 2y + z = 4
⎪
Solve the system of equations ⎨ 2x + y + 4z = 2 .
⎪ 3x − 6y + 3z = 7
⎩
Ex. 5
⎧ x + 2y − 7z = −4
⎪
Solve the system of equations ⎨
.
2x + 3y + z = 5
⎪ 3x + 7y − 36z = −25
⎩
Square Systems
Nonsquare Systems
Ex. 6
⎧⎪ x − y + 4z = 3
Solve the system of equations ⎨
.
−z=0
⎩⎪ 4x
III. Application: Curve Fitting
Ex. 6
Find the equation of the parabola y = ax 2 + bx + c that passes through (0, 0), (3, -3) and (6, 0).