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Unformatted text preview: Section 13.5 Equations of Lines and Planes Generalizing Linear Equations One of the main aspects of single variable calculus was approximating graphs of functions by lines - specifically, tangent lines. In multivari- able calculus, we generalize this idea to the graphs of multivariable functions. In order to do this, first we need to be able to write down the equations for lines in 3-space, and then we shall examine how to write down equations for planes. 1. Equations of Lines In 2-space, the equation for a line is simply a linear equation involv- ing two variables. In 3-space however, this is no longer the case as illustrated by the following example. Example 1.1. Suppose y = x . ( i ) Draw the graph of this equation in 2-space. ( ii ) Draw the graph of this equation in 3-space.-1.0-0.5 0.0 y-2 1.0 0.5-1 0.5 x 0.0 1 2-0.5-1.0 1.0 Note that this is a plane, not a line! 1 2 Instead of just using algebraic equations as we did in 2-space, we need a new way to represent lines in 3-space. There are three different ways we shall use to describe a line, each of which have advantages. 1.1. Vector Equations for Lines. Recall that a line is completely determined by two things - the direction it is pointing and some point on the line. Once these two things are given, the line is completely determined. Since vectors describe a direction, we can use a vector and a point in space to determine a line. We do it as follows: Suppose L is a line passing through the two points P ( a, b, c ) and Q ( x, y, z ). Let vectorv be the displacement vector from P to Q . Define the vector equation vector r ( t ) = vector P + vectorvt where vector P denotes the position vector with head at the point ( a, b, c ). We claim that vector r ( t ) describes the line L completely as t varies over the real numbers. To see this, observe that when vector r (0) = vector P , so the equation vector r ( t ) passes through the point P . Then as t increases or decreases, the equation vector r ( t ) passes through points which lie in the direction determined by vectorv from the point P , and these are exactly the points on L . We summarize: Result 1.2. Suppose L is a line passing through P and Q . If vectorv is the displacement vector from P to Q , then a vector equation for the line L is vector r ( t ) = vector P + vectorvt (notice that this equation is linear in t ). Example 1.3. Write down a vector equation for the line with direction vector 2 vector i + 2 vector j- vector k passing through (2 , 2 , 2). Use it to write down two other points it passes through. We have vector r ( t ) = (2 + 2 t ) vector i + (2 + 2 t ) vector j + (2- t ) vector k To find points on L , we simply plug in values for t . Putting in t = 1 and t = 2, we get the point (3 , 3 , 1) and (4 , 4 , 0)....
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