Section 12.2 : Equations of Lines
4. Determine the intersection point of the line given by \(x = 8 + t\), \(y = 5 + 6t\), \(z = 4 - 2t\) and the line given by \(\vec r\left( t \right) = \left\langle { - 7 + 12t,3 - t,14 + 8t} \right\rangle \) or show that they do not intersect.
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Start SolutionIf the two lines do intersect then they must share a point in common. In other words there must be some value, say \(t = {t_1}\), and some (probably) different value, say \(t = {t_2}\), so that if we plug \({t_1}\) into the equation of the first line and if we plug \({t_2}\) into the equation of the second line we will get the same \(x\), \(y\) and \(z\) coordinates.
Show Step 2This means that we can set up the following system of equations.
\[\begin{align*}8 + {t_1} & = - 7 + 12{t_2}\\ 5 + 6{t_1} & = 3 - {t_2}\\ 4 - 2{t_1} & = 14 + 8{t_2}\end{align*}\]If this system of equations has a solution then the lines will intersect and if it doesn’t have a solution then the lines will not intersect.
Show Step 3Solving a system of equations with more equations than unknowns is probably not something that you’ve run into all that often to this point. The basic process is pretty much the same however with a couple of minor (but very important) differences.
Start off by picking any two of the equations (so we now have two equations and two unknowns) and solve that system. For this problem let’s just take the first two equations. We’ll worry about the third equation eventually.
Solving a system of two equations and two unknowns is something everyone should be familiar with at this point so we’ll not put in any real explanation to the solution work below.
\[\begin{align*}{t_1} = - 15 + 12{t_2} & \hspace{0.2in} \to \hspace{0.2in} & 5 + 6\left( { - 15 + 12{t_2}} \right) & = 3 - {t_2}\\ & & - 85 + 72{t_2} & = 3 - {t_2}\\ & & 73{t_2} & = 88 & \to \hspace{0.2in} {t_2} & = \frac{{88}}{{73}}\\ & & & & {t_1} & = - 15 + 12\left( {\frac{{88}}{{73}}} \right) = - \frac{{39}}{{73}}\end{align*}\] Show Step 4Okay, this is a somewhat “messy” solution, but they will often be that way so we shouldn’t get too excited about it!
Now, recall that to get this solution we used the first two equations. What this means is that if we use this value of \({t_1}\) and \({t_2}\) in the equations of the first and second lines respectively then the \(x\) and \(y\) coordinates will be the same (remember we used the \(x\) and \(y\) equations to find this solution….).
At this point we need to recall that we did have a third equation that also needs to be satisfied at these values of \(t\). In other words, we need to plug \({t_1} = - \frac{{39}}{{73}}\) and \({t_2} = \frac{{88}}{{73}}\) into the third equation and see what we get. Doing this gives,
\[\frac{{370}}{{73}} = 4 - 2\left( { - \frac{{39}}{{73}}} \right) \ne 14 + 8\left( {\frac{{88}}{{73}}} \right) = \frac{{1726}}{{73}}\]Okay, the two sides are not the same. Just what does this mean? In terms of systems of equations it means that \({t_1} = - \frac{{39}}{{73}}\) and \({t_2} = \frac{{88}}{{73}}\) are NOT a solution to the system of equations in Step 2. Had they been a solution then we would have gotten the same number from both sides.
In terms of whether or not the lines intersect we need to only recall that the third equation corresponds to the \(z\) coordinates of the lines. We know that at \({t_1} = - \frac{{39}}{{73}}\) and \({t_2} = \frac{{88}}{{73}}\) the two lines will have the same \(x\) and \(y\) coordinates (since they came from solving the first two equations). However, we’ve just shown that they will not give the same \(z\) coordinate.
In other words, there are no values of \({t_1}\) and \({t_2}\) for which the two lines will have the same \(x\), \(y\) and \(z\) coordinates. Hence, we can now say that the two lines do not intersect.
Before leaving this problem let’s note that it doesn’t matter which two equations we solve in Step 3. Different sets of equations will lead to different values of \({t_1}\) and \({t_2}\) but they will still not satisfy the remaining equation for this problem and we will get the same result of the lines not intersecting.