Explanation:

Let’s break this down, point by point.

- We have three position vectors:

P: a?–?2b?+?3c

Q: 2a?+?3b?–?4c

R: –7b?+?10c

- First, always check if they’re collinear. That means the vectors from one point to the others are proportional—basically, all the points line up on a single straight line.

- If not, then maybe they’re coplanar. That’s less restrictive: any three vectors in three-dimensional space (as long as they’re not all the same line) will always sit in some plane together.

- In fact, any three points in space will always be coplanar, because you can always draw a plane through three points.

- If you see the option “non-collinear,” that just means the vectors don’t line up but might be coplanar—so, not on a straight line, but sitting on the same flat surface.

- “Neither” would mean the points don’t even share a plane, but that can’t happen with only three points.

Let’s do a quick calculation:

Let’s check if vectors PQ and PR are proportional.

PQ = Q - P = (2a?+?3b?–?4c) – (a?–?2b?+?3c)

PQ = (2a – a) + (3b – (–2b)) + (–4c – 3c)

= a + 5b – 7c

PR = R – P = (–7b?+?10c) – (a?–?2b?+?3c)

= –a + (–7b – (–2b)) + (10c – 3c)

= –a – 5b + 7c

You can see that PR is exactly the negative of PQ. That means all three points fall on the same line. Collinear.

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Option 1: Collinear

This is the correct answer. The vectors are proportional.

Option 2: Coplanar

This is always true for three points, but collinear is a stricter answer.

Option 3: Non-collinear

Nope, the vectors clearly are proportional, so they are absolutely collinear.

Option 4: Neither

Nope, not possible with three points.

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