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Real world problem: Somebody stays on an inclined surface. In which direction will they fall?

Description:
I place on the ground a wooden plank (W). It is parallel to the ground (G) (the ground is the reference).
The size of the plank is irrelevant. The plank has no thickness.

I stick 3 bricks under the plank, at 3 random places (not necessary under the corners of the plank).
One's brick height (z) brick is 10cm high, one's is 20 and one's is 30 (or any random size you like). This will obviously tilt the plank. Depending on the size of the bricks, the plank could be tilted only on Y axis (towards north-south), only on the X axis (east-west) or both axis.
We know the coordinates of the bricks (P1, P2, P3).

$$ \cases{ p_1 =(x_1,y_1,z_1)\\ p_2 = (x_2,y_2,z_2)\\ p_3 =(x_3,y_3,z_3) } $$

(Clarification: X, Y relative to the bottom-left corner of the plank (which we make 0,0), not to the ground.)

Now, I drill a hole into the plank (random point) and put a stick through the hole, into the ground. The stick will be perfectly perpendicular TO THE GROUND.

Question: Imagine somebody placed the briks under the plank and now that person sits on the plank. Without looking at the plank, I need to tell in which direction he is looking (north, south, east, west, or combination of those).
(The north, south, east, west are made up names. They do not have to be the real Earth coordinates. I just try to exemplify that the plank will have an inclination towards one of the 4 directions. )

And I also need to know two angles:

  • the angle between the stick and one edge of the plank and
  • the angle between the stick and the other edge (perpendicular to the first edge).

I will upload a sketch also if the description is not enough.

GoUkraineGo
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  • What have you tried, to get started? – 311411 Aug 26 '21 at 13:58
  • I solved the problem when the bricks are exactly under the corners. In this case, it is really easy because the bricks (P1, P2, P3) will form two perpendicular lines. I calculate the angle to a line and the angle to the other line. This gives me the direction of the inclination, Easy-peasy. – GoUkraineGo Aug 26 '21 at 14:00

1 Answers1

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if you have the coordinates of your three points, its easy to find the equation of the plain of the plank, therefore you know the normal vector of the plain, you do not need the stick to know the inclination of the plain.

trula
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  • Considering the 3 bricks (P1= 0,0,1, P2= 0,10,0, P3= 10,0,2), can you please help me calculate the orientation/angle between the stick and the plane, on the x AND y axis? – GoUkraineGo Aug 26 '21 at 14:22
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    @Yes; this inclination has been computed in response to the OP's previous question, which leads to this follow-up question. – ryang Aug 26 '21 at 15:26