Consider the plane that contains the three points

p=[3,6,1],

q=[7,2,2]

r=[9,2,1].

Using p as your reference point, compute n = (q-p) x (r-p).

Then, using n and the same reference point p, write the equation of the plane in the form

ax + by + cz = 1

Enter the value of a. Give your answer correct to 2 decimal places.

Note: For the rare scenario that your answer is undefined (i.e. requires a division by 0), enter -999 as your answer. Submit a mark query only if the correct answer shows "NaN" (Not a Number).

Question

Consider the plane that contains the three points

p=[3,6,1],

q=[7,2,2]

r=[9,2,1].

Using p as your reference point, compute n = (q-p) x (r-p).

Then, using n and the same reference point p, write the equation of the plane in the form

ax + by + cz = 1

Enter the value of a. Give your answer correct to 2 decimal places.

Note: For the rare scenario that your answer is undefined (i.e. requires a division by 0), enter -999 as your answer. Submit a mark query only if the correct answer shows "NaN" (Not a Number).

BlackTom AI · Accepted Answer

Compute vectors q - p and r - p, then form their cross product to get the normal vector n.

1) Relevant concepts
- The normal vector n to a plane through p, q, r is n = (q - p) × (r - p).
- The plane equation using reference point p is n · (x - p) = 0, which expands to n·x = n·p. To get ax + by + cz = 1, divide both sides by the scalar n·p.

2) Step-by-step computation
- q - p = [7, 2, 2] - [3, 6, 1] = [4, -4, 1].
- r - p = [9, 2, 1] - [3, 6, 1] = [6, -4, 0].

- Cross product n = (q - p) × (r - p).
  Using component formula:
  n_x = (-4)*0 - 1*(-4) = 0 + 4 = 4
  n_y = 1*6 - 4*0 = 6 - 0 = 6
  n_z = 4*(-4) - (-4)*6 = -16 + 24 = 8
  So n = [4, 6, 8].

- Plane equation: n · (x - p) = 0 ⇒ n·x = n·p.
  Compute n·p = 4*3 + 6*6 + 8*1 = 12 + 36 + 8 = 56.

Hence 4x + 6y + 8z = 56.

- Put into form ax + by + cz = 1 by dividing both sides by 56:
  a = 4/56 = 1/14 ≈ 0.0714285714.

3) Rounding and final numeric value
- Rounded to two decimal places: a = 0.07.

4) Why this is correct
- The cross product gives a correct normal direction; n·(x - p)=0 enforces that every vector from p to any point (x,y,z) on the plane is orthogonal to n. Dividing by n·p yields the requested right-hand side of 1 without changing the plane. 4/56 simplifies exactly to 1/14, which is ≈0.0714, so 0.07 is the two-decimal rounding.

5) Common incorrect approaches
- Using the wrong sign for the cross product components or computing n·p incorrectly would produce wrong coefficients; dividing by the wrong scalar (e.g., dividing by 4 instead of 56) would give incorrect a values like 1.00 or 0.14. None of those are consistent with the correct algebra above.

Final confirmation: the value of a rounded to two decimals is 0.07, matching the provided answer.

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