SOLUTION: The lines l and m have vector equations r = 2i-j+4k+s(i+j- k) and r= -2i + 2j+ k+ t(-2i+j+ k) respectively・ The point P lies on l and the point Q has position vector 2i

SOLUTION: The lines l and m have vector equations r = 2i-j+4k+s(i+j- k) and r= -2i + 2j+ k+ t(-2i+j+ k) respectively・ The point P lies on l and the point Q has position vector 2i - k. (

Question 1207408: The lines l and m have vector equations
r = 2i-j+4k+s(i+j- k) and r= -2i + 2j+ k+ t(-2i+j+ k) respectively・
The point P lies on l and the point Q has position vector 2i - k.
(i) Given that the line PQ is perpendicular to l, find the position vector of P.
(ii) Verify that Q lies on m and that PQ is perpendicular to m.
for part i) i have done it and got the answer for position vector P = 4i + j + 2k
im having problem with part ii)
Answer by Timnewman(323) (Show Source): You can put this solution on YOUR website!
Hello, here is the solution to your vector problem.
=≠====≠==================
For part (i), you mentioned you've already found the position vector of P, so let's move on to part (ii).
To verify that Q lies on m, you can substitute the position vector of Q (2i - k) into the vector equation of line m:
r = -2i + 2j + k + t(-2i + j + k)
Substitute r = 2i - k:
2i - k = -2i + 2j + k + t(-2i + j + k)
Simplify and solve for t:
t = 1
This means Q lies on line m.
To verify that PQ is perpendicular to m, you can find the dot product of the vectors PQ and the direction vector of line m (-2i + j + k).
First, find the vector PQ:
PQ = Q - P (use the position vector of P you found in part i)
Then, find the dot product:
(PQ) · (-2i + j + k) = 0
If the dot product is zero, it means PQ is perpendicular to line m.
Please share the position vector of P you found in part (i), and I can help you with the remaining calculations!
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Incase you need more clarification, I am open to teaching online.
Thank.
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