Question

Describe and sketch the surface which has the equation\(x + y = 2{\rm{ in }}{\mathbb{R}^3}\).

Short answer

The equation\(x + y = 2{\rm{ in }}{\mathbb{R}^3}\) represents the vertical plane that intersects the\(xy - \) plane in the line\({\rm{y = 2 - x}},{\rm{z = 0}}.\)

Step-by-step solution

Three-dimensional coordinate system.

\({\mathbb{R}^3}\)Is the three-dimensional coordinate system which contains\(x,y\)and\(z - \)coordinates.

The equation of the surface is also written as\({\rm{y = 2 - x}}.\)

The equation\({\rm{y = 2 - x}}.\)in\({\mathbb{R}^3}\)represents the set\(\{ (x,2 - x,z)\mid x \in \mathbb{R},z \in \mathbb{R}\} ,\) which is the set of all points in\({\mathbb{R}^3}\) such that the sum of\(x - \) and\(y - \) coordinates is\(2.\)

Therefore, the surface\({\rm{y = 2 - x}}.\)in\({\mathbb{R}^3}\)forms a vertical plane that intersects the\(xy - \) plane in the line\({\rm{y = 2 - x}},{\rm{z = 0}}.\)

Use the explanation to sketch the surface.

From the equation, sketch the surface which has the equation\({\rm{y = 2 - x}}.\)in\({\mathbb{R}^3}\)as shown in Figure 1.

Thus, the surface which has the equation\({\rm{y = 2 - x}}.\)in\({\mathbb{R}^3}\)is described and sketched.