7.3 3-D Notes Honors Precalculus Date: Adapted from 11.1 & 11.4

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1 73 3-D Notes Honors Precalculus Date: Adapted from 111 & 114 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 z- axis 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

2 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

3 73 Notes Honors Precalculus Date: I Multivariable Linear Systems: Row-Echelon Form & Back Substitution System of Three Linear Equations in Three Variables: Row-Echelon Form x 2y + 3z = 9 x + 3y = 4 2x 5y + 5z = 17 Equivalent System in Row- Echelon Form: x 2y + 3z = 9 y + 3z = 5 z = 2 Ordered Triple Ex 1 Use back-substitution in row-echelon form to solve this system of linear equations 2x y + 5z = 22 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

4 Ex 2 2x + y = 90 x + 2y = 90 Ex 3 x y + z = 4 x + 3y 2z = 3 3x + 2y + 2z = 6 Ex 4 x 2y + z = 4 2x + y + 4z = 2 3x 6y + 3z = 7

5 Ex 5 x + 2y 7z = 4 2x + 3y + z = 5 3x + 7y 36z = 25 Square Systems Nonsquare Systems Ex 6 x y + 4z = 3 4x z = 0 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)