Question

Consider 3 Points A(1,0,2),B(1,−1,2),C(5,4,−1) (A) Find The Plane P(X,Y,Z) Passing Through The Three Points A,B,C Above.Show That Three point above .

Answer 1

To find the equation of the plane passing through three given points, we can find the normal vector of the plane and use one of the points to determine the value of a constant in the equation. In this case, the equation of the plane P passing through the points A(1,0,2), B(1,-1,2), and C(5,4,-1) is x + z - 2 = 0.

Step-by-step explanation:

To find the equation of the plane passing through points A(1,0,2), B(1,-1,2), and C(5,4,-1), we can use the formula for the equation of a plane, which is Ax + By + Cz + D = 0.

First, let's find the normal vector of the plane by taking the cross product of the vectors formed by the points AB and AC:

1. Vector AB = B - A = (1, -1, 2) - (1, 0, 2) = (0, -1, 0).
2. Vector AC = C - A = (5, 4, -1) - (1, 0, 2) = (4, 4, -3).
3. Normal vector N = AB x AC = (0, -1, 0) x (4, 4, -3) = (-4, 0, -4).

Now that we have the normal vector, we can substitute one of the given points (A, B, or C) into the equation Ax + By + Cz + D = 0 to find the value of D:

1. Using point A(1, 0, 2): 1*(-4) + 0*0 + 2*(-4) + D = 0, which gives D = 8.

Therefore, the equation of the plane P passing through the three points A, B, and C is -4x - 4z + 8 = 0, which simplifies to x + z - 2 = 0.