Find the z-coordinate of the point on XOZ plane divides the join of (5, -3, -2) and (1, 2, -2) .

Options

A. 2
B. `-2`
C. 0
D. `(13)/(5)`

Answer

To find the z-coordinate of the point on the XOZ plane that divides the line segment joining the points (5, -3, -2) and (1, 2, -2), we can follow these steps:

### Step 1: Identify the Points
We have two points:
- Point A: \( A(5, -3, -2) \)
- Point B: \( B(1, 2, -2) \)

### Step 2: Write the Vector Equation of the Line
The vector equation of the line joining points A and B can be expressed as:
\[ \mathbf{r} = \mathbf{a} + \lambda \mathbf{d} \]
where \( \mathbf{a} \) is the position vector of point A, and \( \mathbf{d} \) is the direction vector from A to B.

### Step 3: Calculate the Direction Vector
The direction vector \( \mathbf{d} \) can be calculated as:
\[ \mathbf{d} = B - A = (1 - 5, 2 - (-3), -2 - (-2)) = (-4, 5, 0) \]

### Step 4: Write the Vector Equation
The vector equation of the line can be written as:
\[ \mathbf{r} = (5, -3, -2) + \lambda (-4, 5, 0) \]
This expands to:
\[ \mathbf{r} = (5 - 4\lambda, -3 + 5\lambda, -2) \]

### Step 5: Determine the Condition for the XOZ Plane
For a point to lie in the XOZ plane, the y-coordinate must be zero. Thus, we set the y-component of the vector equation to zero:
\[ -3 + 5\lambda = 0 \]

### Step 6: Solve for \( \lambda \)
Solving the equation:
\[ 5\lambda = 3 \implies \lambda = \frac{3}{5} \]

### Step 7: Substitute \( \lambda \) Back to Find the Coordinates
Now, substitute \( \lambda = \frac{3}{5} \) back into the vector equation to find the coordinates:
\[ x = 5 - 4\left(\frac{3}{5}\right) = 5 - \frac{12}{5} = \frac{25}{5} - \frac{12}{5} = \frac{13}{5} \]
\[ y = -3 + 5\left(\frac{3}{5}\right) = -3 + 3 = 0 \]
\[ z = -2 \]

### Step 8: Write the Final Coordinates
Thus, the coordinates of the point that divides the line segment in the XOZ plane are:
\[ \left(\frac{13}{5}, 0, -2\right) \]

### Step 9: Identify the z-coordinate
The z-coordinate of this point is:
\[ \text{z-coordinate} = -2 \]

### Final Answer
Hence, the z-coordinate is:
\[ \boxed{-2} \]