UNIT 7LINES AND PLANESINTRO: This Unit is about lines and planes: their equations, how they are determined bypoints and vectors, and some additional facts about them. In the discussions of this Unit,x,yandzwill be variables, anda,b,canddwill be constants.
or, equivalently,~v=R-Q, and either of the given points as your pointP.To find the line through the pointP= (p1, p2, p3) parallel to the linex=d+at, y=e+bt, z=f+ct, realizing that the coefficients oftgive the vector~v, use~v=< a, b, c >and of course the specified point.To find the line through the pointP= (p1, p2, p3) perpendicular to the planeax+by+cz=d,we need a fact proved in the next section: the vector~n=< a, b, c >is perpendicular to theplane. But this is just the direction we need, so use~v=< a, b, c >and again the specified point.Example: All examples use the same strategy – find a vector~vin the direction of the line,and then use it and one point to write the equation of the line. For example, to find the linethroughP= (1,2,3) parallel to the linex= 4 + 3ty= 1 +tz= 2-t.one sees that~v=<3,1,-1>is the direction of the given line, whence it will also be used inthe answer, and the desired line isx= 1 + 3ty= 2 +tz= 3-tExample: Find the equations of a line through the pointsP= (3,-2,