Describe the surface in $\mathbb{R}^3$ represented by the equation $x + y = 3$ .

(a) This is the set $\{x, 3 - x, z)| x \in \mathbb{R}, z \in \mathbb{R}\}$ which is a vertical plane that intersects the $xz$ -plane in the line $y = 3 - x, z = 0$ .

(b) This is the set $\{x, 3 - x, z)| x \in \mathbb{R}, z \in \mathbb{R}\}$ which is a horizontal plane that intersects the $xy$ -plane in the line $y = 3 - x, z = 0$ .

(c) This is the set $\{x, 3 - x, z)| x \in \mathbb{R}, z \in \mathbb{R}\}$ which is a vertical plane that intersects the $xy$ -plane in the line $y = 3 - x, z = 0$ .

(d) This is the set $\{x, 3 - x, z)| x \in \mathbb{R}, z \in \mathbb{R}\}$ which is a horizontal plane that intersects the $xz$ -plane in the line $y = 3 - x, z = 0$ .

(e) This is the set $\{x, y, 3 - x - y)| x \in \mathbb{R}, y \in \mathbb{R}\}$ which is a vertical plane that intersects the $xy$ -plane in the line $y = 3 - x, z = 0$ .

Question :

Describe the surface in $\mathbb{r}^3$ represented by the equation $x + y = 3$ .

(a) this is the set $\{x, 3 - x, z)| x \in \mathbb{r}, z \in \mathbb{r}\}$ which is a vertical plane that intersects the $xz$ -plane in the line $y = 3 - x, z = 0$ .
(b) this is the set $\{x, 3 - x, z)| x \in \mathbb{r}, z \in \mathbb{r}\}$ which is a horizontal plane that intersects the $xy$ -plane in the line $y = 3 - x, z = 0$ .
(c) this is the set $\{x, 3 - x, z)| x \in \mathbb{r}, z \in \mathbb{r}\}$ which is a vertical plane that intersects the $xy$ -plane in the line $y = 3 - x, z = 0$ .
(d) this is the set $\{x, 3 - x, z)| x \in \mathbb{r}, z \in \mathbb{r}\}$ which is a horizontal plane that intersects the $xz$ -plane in the line $y = 3 - x, z = 0$ .
(e) this is the set $\{x, y, 3 - x - y)| x \in \mathbb{r}, y \in \mathbb{r}\}$ which is a vertical plane that intersects the $xy$ -plane in the line $y = 3 - x, z = 0$ .

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Solution:

Neetesh Kumar | December 19, 2024

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This is the solution to Math 1C
Assignment: 12.1 Question Number 6
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Step-by-step solution:

The equation $x + y = 3$ represents a vertical plane in $\mathbb{R}^3$ .

Step 1: The general representation of this plane is:

$\{(x, 3 - x, z) | x \in \mathbb{R}, z \in \mathbb{R}\}.$

Step 2: The plane:

Intersects the $xy$ -plane when $z = 0$ . Substituting $z = 0$ into $x + y = 3$ , we get: $y = 3 - x, \quad z = 0.$
Intersects the $xz$ -plane when $y = 0$ . Substituting $y = 0$ into $x + y = 3$ , we get: $x = 3, \quad y = 0.$

Thus, this is a vertical plane that passes through the $xy$ -plane and $xz$ -plane.

Final Answer:

The correct description is:

(c) $\boxed{\text{This is the set} \space \{(x, 3 - x, z) | x \in \mathbb{R}, z \in \mathbb{R}\} \space \text{which is a vertical plane that intersects the} \space xy \text{-plane in the line} \space y = 3 - x, z = 0}$

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