Question 1207408
Hello, here is the solution to your vector problem.
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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.