Question

To describe: The representation of equation \(x = 5\) in \({\mathbb{R}^3}\).

Short answer

The equation \(x = 5\) in \({\mathbb{R}^3}\) represents the vertical plane that is parallel to the \(yz\)-plane and it locates 5 units below the \(yz\)-plane.

Step-by-step solution

Concept of \({\mathbb{R}^3}\)

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

Represent the given equation in \({\mathbb{R}^3}\)

The equation \(x = 5\) in \({\mathbb{R}^3}\) represents the set \(\{ (x,y,z)\mid x = 5\} \), which is the set of all points in \({\mathbb{R}^3}\) whose \(x\) coordinate is \(5\) and \(y\), \(z\)-coordinates are any values.

The equation \(x = 5\) in \({\mathbb{R}^3}\) is sketched as shown in Figure 1.

From Figure 1, the equation \(x = 5\) in \({\mathbb{R}^3}\) represents the vertical plane and it is parallel to the \(yz\)-plane. It locates \(5\) units below the \(yz\)-plane.

Thus, the equation \(x = 5\) in \({\mathbb{R}^3}\) is described.