Page 1 Summary:

This page covers solutions to problems involving points and planes in 3D space. It begins with finding the coordinates of point P(4,1,4,1,5) on a plane given by the equation x+y+2z=6. The solution involves setting up and solving a system of equations. The page also includes calculations of distances between points using the distance formula in three dimensions.

Example: For point P(4,1,4,1,5), the solution shows how to set up the equation 1(4+t) + 1(4+t) + 2(5+2t) = 6 and solve for t to find the coordinates.

Vocabulary: The distance formula in 3D space is given as |PD| = √(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)².

Page 3 Summary:

Page 3 focuses on solving problems related to the equation of a plane 3y+4z=0. It includes finding coordinates of multiple points on this plane and calculating distances between these points. The solutions demonstrate how to use vector equations and solve for the parameter t to find point coordinates.

Example: For point A(3,1,-1,1,7), the solution shows how to set up the equation 3(-1+3t) + 4(7+4t) = 0 and solve for t to find the coordinates.

Vocabulary: The vector equation of a line is often written in the form r = a + tb, where a is a point on the line, b is a direction vector, and t is a parameter.