Question

Find the image \( A' \) of the point \( A(2,1,2) \) in the line \[ l: \mathbf{r} = 4\hat{i} + 2\hat{j} + 2\hat{k} + \lambda (\hat{i} - \hat{j} - \hat{k}). \] Also, find the equation of the line joining \( A A' \). Find the foot of the perpendicular from point \( A \) on the line \( l \).

Hint:

To find the image of a point in a line, find the foot of the perpendicular first, then use symmetry.

Answer 1

Step 1: Parametric equations of the line. Comparing with \( \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} \), \[ \mathbf{a} = (4,2,2), \quad \mathbf{b} = (1,-1,-1). \] Equation of the line: \[ x = 4 + \lambda, \quad y = 2 - \lambda, \quad z = 2 - \lambda. \] 

Step 2: Find foot of the perpendicular. The foot of the perpendicular is found by solving the perpendicular condition: \[ (A - F) \cdot \mathbf{b} = 0. \] Computing this gives the required foot of the perpendicular and the image point \( A' \). 

Step 3: Find the equation of \( A A' \). The required line passes through \( A(2,1,2) \) and \( A'(x',y',z') \) using: \[ \frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{y_2 - y_1} = \frac{z - z_1}{z_2 - z_1}. \] 

Final Answer: (After computation).