Answer:
The answers to the questions are;
Yes, the lines L1 and L2 intersect.
The point of intersection Q of the lines L1 and L2 is (1, 1, -11).
Step-by-step explanation:
The direction vector for L1 is (1, -1, -3)
and an equation is
x = -3 + t
y = 5 - t
z = 1 - 3·t
The possible equation for L2 is
x = -2 - 2·k
y = 10 + 6·k
z = -5 + 4·k
For intersection, we have
-3 + t = -2 - 2·k AND 5 - t = 10 + 6·k AND 1 - 3·t = -5 + 4 k
From the first equation, we have
t = 1 - 2·k and substituting in the second equation, we get
5 - (1 - 2·k) = 10 + 6·k
4 + 2·k = 10 + 6·k
6 = -4·k and k = -3/2
From which t = 1 - 2·k = 1 -2×(-3/2) = 1 + 3 = 4
t = 4
We now check if the results for t and k satisfies the third equation as follows
Left Hand Side is 1 - 3·t = 1 - 3×4 = -11
Right Hand Side is -5 + 4 k = -5 + (4×(-3/2)) = -5 -6 = -11
Therefore the results from the first two equations satisfy the third equation and we now look for the intersection point as
x = -3 + t ⇒ -3 + 4 = 1
y = 5 - t ⇒ 5 - 4 = 1
z = 1 - 3·t ⇒ 1 - 3×4 = -11
Giving the intersection point Q as (1, 1, -11).
