Find the equation of line through the points P(1, 2, 3) and Q(3, 4, 5) in Cartesian form.

Answer

To find the equation of the line passing through the points P(1, 2, 3) and Q(3, 4, 5) in Cartesian form, we can follow these steps:

### Step 1: Identify the coordinates of the points
Let the coordinates of point P be \( P(x_1, y_1, z_1) = (1, 2, 3) \) and the coordinates of point Q be \( Q(x_2, y_2, z_2) = (3, 4, 5) \).

### Step 2: Write down the formula for the equation of the line
The equation of the line in three-dimensional space through two points can be expressed as:
\[ \frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{y_2 - y_1} = \frac{z - z_1}{z_2 - z_1} \]

### Step 3: Substitute the coordinates into the formula
Substituting the coordinates of points P and Q into the equation:
- \( x_1 = 1, y_1 = 2, z_1 = 3 \)
- \( x_2 = 3, y_2 = 4, z_2 = 5 \)

We get:
\[ \frac{x - 1}{3 - 1} = \frac{y - 2}{4 - 2} = \frac{z - 3}{5 - 3} \]

### Step 4: Simplify the equation
Calculating the differences:
- \( 3 - 1 = 2 \)
- \( 4 - 2 = 2 \)
- \( 5 - 3 = 2 \)

Thus, the equation simplifies to:
\[ \frac{x - 1}{2} = \frac{y - 2}{2} = \frac{z - 3}{2} \]

### Step 5: Write in Cartesian form
We can express this in Cartesian form by eliminating the fractions:
\[ x - 1 = 2k, \quad y - 2 = 2k, \quad z - 3 = 2k \]
where \( k \) is a parameter. Rearranging gives:
\[ x - 1 = y - 2 = z - 3 \]

### Final Answer
The equation of the line passing through points P(1, 2, 3) and Q(3, 4, 5) in Cartesian form is:
\[ x - 1 = y - 2 = z - 3 \]

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