Q: Let L1 be the line passing through the points
Q1=(−3, 5, −4) and Q2=(−9, −1, 2).
Find a value of k so the line L2 passing through the point P1 = P1(−1, 11, k) with direction vector
→d=[1, −3, −3]T intersects with L1.
k = ?
Can someone please show me the answer and how they got it. This is what I have done so far:
v1 = Q2 – Q1 = (-6, -6, 6)
L1:
x = -3 -6t
y = 5 – 6t
z = -4 + 6t
L2
passing through the point P1 = P1(−1, 11, k) with direction
vector →d=[1, −3, −3]:
x = -1 – 1
y = 11 – 3
z = k -3
L₁ will intersect L₂ IF the simultaneous equation admit a solution
-3 -6t = 1 – 1
11 – 3 = 5 – 6t
-4 + 6t = k -3
**On a sidenote does anyone else have difficulty with really long calculations? I've understood the concept for similiar questions but when it comes to the computation my brain just completely shuts down and I make lots of silly mistakes 🙁 **
Best Answer
You're on the right track. Your first line has equations: $$ \begin{bmatrix} x\\y\\z \end{bmatrix} = \begin{bmatrix} -3\\5\\-4 \end{bmatrix} + t \begin{bmatrix} -6\\-6\\6 \end{bmatrix} $$
You didn't have any parameter on your second line so you are missing some variables in your system. The second line has equation: $$ \begin{bmatrix} x\\y\\z \end{bmatrix} = \begin{bmatrix} -1\\11\\k \end{bmatrix} + s \begin{bmatrix} 1\\-3\\-3 \end{bmatrix} $$
If you now simultaneously solve these you should be able to find $k$.
As for tips for long calculations, I think a lot of it boils down to practice. As you do more problems you will build intuition for the big steps you need to take to solve different types of problems, and also for things which might have gone wrong if you get stuck.
For instance, the system you obtained has no solution. However, if you think about the lines in real life it seems like there should be a solution. So you can then go back and try to look at why your system didn't have a solution.
Sometimes it also helps to slow down and really write out what you are doing at each step. I also make a lot of mistakes even when I have an idea how to solve the problem, and sometimes it just takes me sitting down and being very deliberate about each step of the solution to eliminate all my silly mistakes.