Final answer:
The Cartesian equation of the plane containing points P=(1,1,1), Q=(0,1,0), and R=(0,0,1) is found by determining the normal vector through the cross product of vectors PQ and PR, which results in the equation y + z = 2.
Step-by-step explanation:
The question asks us to find the Cartesian equation of the plane containing the points P=(1,1,1), Q=(0,1,0), and R=(0,0,1). To find this equation, we need to use two vectors that lie in the plane. We can use vector PQ and vector PR to construct a normal vector by finding the cross product. The cross product of two vectors PQ = Q - P and PR = R - P will give us a vector that is perpendicular to the plane.
First, we calculate the vectors PQ and PR:
Second, we find the cross product of these vectors (PQ x PR) to get the normal vector N:
Now, we can use the normal vector N to write the plane's equation as:
N · (x - x0, y - y0, z - z0) = 0
Where (x0, y0, z0) is a point on the plane, which can be P, Q, or R. Using point P, we get:
0(x - 1) -1(y - 1) -1(z - 1) = 0
or simply,
y + z = 2.
This is the Cartesian equation of the plane.