Question

The image of the point with position vector \[ \mathbf{i} + 3 \mathbf{k} \] in the plane \( \mathbf{r} \cdot (\mathbf{i} + \mathbf{j} + \mathbf{k}) = 1 \) is:

• A. \( \mathbf{i} - 2\mathbf{j} + \mathbf{k} \)
• B. \( \mathbf{i} + 2\mathbf{j} - \mathbf{k} \)
• C. \( -\mathbf{i} - 2\mathbf{j} + \mathbf{k} \)
• D. None of these

Hint:

When finding the image of a point across a plane, use the reflection formula to obtain the coordinates of the reflected point.

Answer 1

The Correct Option is C

Step 1: Set up the equation for the plane.
The equation of the plane is: \[ \mathbf{r} \cdot (\mathbf{i} + \mathbf{j} + \mathbf{k}) = 1 \] We are given that the position vector of the point is \( \mathbf{i} + 3\mathbf{k} \), which corresponds to the point \( (1, 0, 3) \).

Step 2: Use the reflection formula.
To find the image of the point in the plane, we use the formula for the reflection of a point across a plane. If the point \( P(x_1, y_1, z_1) \) is reflected across a plane with normal vector \( \mathbf{n} = (a, b, c) \), the image point \( P' \) is given by: \[ \mathbf{r}' = \mathbf{r} - 2 \left( \frac{\mathbf{r} \cdot \mathbf{n} - d}{\|\mathbf{n}\|^2} \right) \mathbf{n} \] Here, \( \mathbf{n} = (1, 1, 1) \), and \( d = 1 \).

Step 3: Calculate the dot product.
The dot product \( \mathbf{r} \cdot \mathbf{n} \) is: \[ (1, 0, 3) \cdot (1, 1, 1) = 1 + 0 + 3 = 4 \]

Step 4: Find the scalar for the reflection.
Now, calculate the scalar: \[ \frac{\mathbf{r} \cdot \mathbf{n} - d}{\|\mathbf{n}\|^2} = \frac{4 - 1}{1^2 + 1^2 + 1^2} = \frac{3}{3} = 1 \]

Step 5: Calculate the image vector.
Using the formula for reflection: \[ \mathbf{r}' = (1, 0, 3) - 2(1)(1, 1, 1) = (1, 0, 3) - (2, 2, 2) = (-1, -2, 1) \] Thus, the image of the point is \( -\mathbf{i} - 2\mathbf{j} + \mathbf{k} \), corresponding to option (C).