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P is a point and Q a point on a given line: find mu so that PQ is parallel to a given plane

The same question, asked 3 times: 3 May 2023 S1*, 11 May 2023 S2*, 15 May 2023 S1. Every wording below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    3 May 2023 S1

    146
    Let P(2,3,6)P(2,3,6) be a point in space and Q be a point on the line r⃗=(i^−j^+2k^)+μ(−3i^+j^+5k^)\vec{r}=(\hat{i}-\hat{j}+2\hat{k})+\mu(-3\hat{i}+\hat{j}+5\hat{k}). Then the value of μ\mu for which vector PQ is parallel to the plane x−4y+4z=1x-4y+4z=1 is

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  2. Question 2

    11 May 2023 S2

    145
    Let P(3,2,6)P(3,2,6) be a point in space and Q be a point on the line r⃗=i^−j^+2k^+μ(−3i^+j^+5k^)\vec{r}=\hat{i}-\hat{j}+2\hat{k}+\mu(-3\hat{i}+\hat{j}+5\hat{k}). Then the value of μ\mu for which the vector PQ→\overrightarrow{PQ} is parallel to the plane x−4y+3z=1x-4y+3z=1 is

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  3. Question 3

    15 May 2023 S1

    138
    Let P(2,1,5) be a point in space and Q be a point on the line r⃗=(i^−j^+2k^)+μ(−3i^+j^+5k^)\vec{r}=(\hat{i}-\hat{j}+2\hat{k})+\mu(-3\hat{i}+\hat{j}+5\hat{k}). Then the value of μ\mu for which the vector PQ is parallel to the plane 3x−y+4z=13x-y+4z=1 is

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