Find The Equation of the plane in $R^3$ known two points p1 , p2 on the plane and the distance between the plane and a third given point p3 is d

Assume $$p1=(x_1,y_1,z_1)$$ , $$p2=(x_2,y_2,z_2)$$ and $$p3=(x_3,y_3,z_3)$$

THIS IS HOW I IMAGINE IT (so it may be wrong)

$$p_1p_2^\rightarrow =(x_2-x_1,y_2-y_1,z_2-z_1)$$ which is a vector on the plane

i tried to get the projection vector of $$p_1p_3^\rightarrow$$ onto $$p_1p_2^\rightarrow$$ then the rejection Which as i think so is the normal vector $$n^\rightarrow$$ according to https://en.wikipedia.org/wiki/Vector_projection

states that the vector rejection is the orthogonal projection of $$p_1p_3^\rightarrow$$ onto the plane

vector projection of $$p_1p_3^\rightarrow$$ onto $$p_1p_2^\rightarrow =$$ $$\frac{p_1p_3^\rightarrow.p_1p_2^\rightarrow}{\lVert p_1p_2^\rightarrow\rVert^2}p_1p_2^\rightarrow=v^\rightarrow$$

then the Vector rejection which is the normal vector $$n^\rightarrow$$ =$$p_1p_3^\rightarrow - v^\rightarrow$$

then the equation of the plane is $$n^\rightarrow.((x,y,z)-(x_1,y_1,z_1))=0$$

i don't know if this is right or wrong because as you see i didn't use the distance he gave and i searched a lot but i found nothing

so i need any clarification please

• There is an infinite number of planes that contain the line through $p_1$ and $p_2$. The distance from $p_3$ to this line is equal to the distance from $p_3$ to only one of these planes; for the rest, the distance is less than this.
– amd
Commented Nov 25, 2019 at 23:49

Calling $$p^*$$ the tangent point between the plane and the sphere $$||p-p_3||= d$$, we have the relations

$$\cases{ ||p^*-p_3||^2=d^2\\ (p^*-p_3)\cdot(p_1-p_2) = 0\\ (p^*-p_3)\cdot(p_1-p^*) = 0 }$$

We have three conditions and three unknowns which are the $$p^*$$ components.

Case study

$$\cases{ p_1 = (1,0,1)\\ p_2 = (2,-1,0)\\ p_3 = (-2,1,-2)\\ d = 2 }$$

$$p^* = \left(-\frac{2}{7} \left(5+\sqrt{11}\right),-\frac{3}{7} \left(\sqrt{11}-2\right),\frac{1}{7} \left(\sqrt{11}-9\right)\right)$$

and the plane equation is

$$(p-p^*)\cdot(p^*-p_3)=0,\ \ \ p=(x,y,z)$$

• omg thank you so much Commented Nov 30, 2019 at 0:25