Calculus II - Equations of Lines

Show Mobile Notice

Mobile Notice

You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best viewed in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll/swipe to see them) and some of the menu items will be cut off due to the narrow screen width.

Section 12.2 : Equations of Lines

Back to Problem List

6. Does the line given by $x = 9 + 21t$ , $y = - 7$ , $z = 12 - 11t$ intersect the $xy$ -plane? If so, give the point.

Show All Steps Hide All Steps

Start Solution

If the line intersects the $xy$ -plane there will be a point on the line that is also in the $xy$ ‑plane. Recall as well that any point in the $xy$ -plane will have a $z$ coordinate of $z = 0$ .

Show Step 2

So, to determine if the line intersects the $xy$ -plane all we need to do is set the equation for the $z$ coordinate equal to zero and solve it for $t$ , if that’s possible.

Doing this gives,

$12 - 11t = 0\hspace{0.25in}\hspace{0.25in} \to \hspace{0.25in}\hspace{0.25in}t = \frac{{12}}{{11}}$ Show Step 3

So, we were able to solve for $t$ in this case and so we can now say that the line does intersect the $xy$ -plane.

Show Step 4

All we need to do to finish this off this problem is find the full point of intersection. We can find this simply by plugging $t = \frac{{12}}{{11}}$ into the $x$ and $y$ portions of the equation of the line.

Doing this gives,

$x = 9 + 21\left( {\frac{{12}}{{11}}} \right) = \frac{{351}}{{11}}\hspace{0.25in}\hspace{0.25in}y = - 7$

The point of intersection is then : $\require{bbox} \bbox[2pt,border:1px solid black]{{\left( {\frac{{351}}{{11}}, - 7,0} \right)}}$ .