Finding components of lines intersecting at a point.

by Ella   Last Updated August 09, 2018 21:20 PM

I'm just working on some summer problems so that I can be more prepared when I go into my class in the fall. I found a website full of problems of the content we will be learning but it doesn't have the answers. I need a little guidance on how to do this problem. Here is the problem:

Consider the lines $L1$ and $L2$ with equations:

$L_1 : r = (11, 8, 2) + s(4, 3, -1)$
$L_2 : r = (1, 1,-7) + t(2, 1, 11)$
The lines intersect at point $P$.

a. Find the coordinates of $P$.
Would we use a formula for this? For example: $\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2}$.

b. Show that the lines are perpendicular.
Isn’t there also a formula to find if lines are perpendicular?

c. The point $Q(7,5,3)$ lies on $L_1$. The point $R$ is the reflection of $Q$ in the line $L_2$.

Find the coordinates of $R$.
If $R$ is the reflection of $Q$, wouldn’t it just be the opposite? Like $-7$?

Answers 1

Ideas to get the answer

Part a):

You need to solve the system

\begin{align} 11+4s&=1+2t\\ 8+3s&=1+t\\ 2-s&=-7+11t \end{align}

Part b):

The lines are perpendicular if and only if its directions are perpendicular. That is, if and only if $$(4,3,-1)\cdot (2,1,11)=0.$$

August 09, 2018 21:00 PM

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