Question

Describe the surface in double-struck r3 represented by the equation x + y = 3. this is the set (x, y, 3 − x − y) which is a vertical plane that intersects the xy-plane in the line y = 3 − x, z = 0.

Answer 1

The equation x + y = 3 represents a vertical plane that intersects the xy-plane in the line y = 3 - x, z = 0.

Step-by-step explanation:

The equation x + y = 3 represents a vertical plane in three-dimensional space. To understand this, let's consider the equation in terms of the Cartesian coordinate system. The equation can be rearranged as follows: y = 3 - x. This equation represents a straight line in the xy-plane, where the y-coordinate is equal to 3 minus the x-coordinate. In three-dimensional space, this line intersects the xy-plane at the line y = 3 - x, with z = 0.

Answer 2

First, note that x+y=3 is a linear in x and y, and independent of z. So, this is just a line in the xy-plane, extended into and out of the plane. Since we have the line, we have all the points we need;

the equation in R3 is a plane parallel to the z-axis and contains the line
y = 3 - x
The plan sketched below.