The problem asks to find the symmetric equations of the line of intersection of two given planes. The planes are given by their equations: $x + y - z = 2$ $3x - 2y + z = 3$

1. Problem Description

The problem asks to find the symmetric equations of the line of intersection of two given planes. The planes are given by their equations:

$x + y - z = 2$

$3x - 2y + z = 3$

2. Solution Steps

First, we need to find the direction vector of the line of intersection. The direction vector $\vec{v}$ is orthogonal to the normal vectors of both planes. The normal vectors are $\vec{n_1} = <1, 1, -1>$ and $\vec{n_2} = <3, -2, 1>$. We find the direction vector by taking the cross product of the normal vectors.

$\vec{v} = \vec{n_1} \times \vec{n_2} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ 1 & 1 & -1 \\ 3 & -2 & 1 \end{vmatrix} = (1 - 2)\vec{i} - (1 - (-3))\vec{j} + (-2 - 3)\vec{k} = -1\vec{i} - 4\vec{j} - 5\vec{k}$

So, $\vec{v} = <-1, -4, -5>$.

Next, we need to find a point on the line of intersection. We can do this by setting one of the variables to a convenient value (e.g., 0) and solving for the other two. Let's set $z = 0$:

$x + y = 2$

$3x - 2y = 3$

Multiply the first equation by 2:

$2x + 2y = 4$

Add this to the second equation:

$5x = 7$, so $x = \frac{7}{5}$.

Substitute $x = \frac{7}{5}$ into $x + y = 2$:

$\frac{7}{5} + y = 2$

$y = 2 - \frac{7}{5} = \frac{10}{5} - \frac{7}{5} = \frac{3}{5}$.

So, a point on the line is $(\frac{7}{5}, \frac{3}{5}, 0)$.

The symmetric equations of the line are given by:

$\frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c}$, where $(x_0, y_0, z_0)$ is a point on the line and $<a, b, c>$ is the direction vector.

So, we have:

$\frac{x - \frac{7}{5}}{-1} = \frac{y - \frac{3}{5}}{-4} = \frac{z - 0}{-5}$

Multiply each fraction by $-1$:

$\frac{x - \frac{7}{5}}{1} = \frac{y - \frac{3}{5}}{4} = \frac{z}{5}$

Multiply each term by 5 to remove fractions:

$\frac{5x - 7}{5} = \frac{5y - 3}{20} = \frac{z}{5}$

$\frac{5x-7}{1} = \frac{5y-3}{4} = z$

$\frac{x - \frac{7}{5}}{1} = \frac{y - \frac{3}{5}}{4} = \frac{z}{5}$

3. Final Answer

$\frac{x - \frac{7}{5}}{1} = \frac{y - \frac{3}{5}}{4} = \frac{z}{5}$