To find the foot of the perpendicular from A to the line l, consider a general point P on l with parameter \(\lambda\), given by:
\(\overrightarrow{OP} = (9 + 3\lambda)\mathbf{i} + (-1 - \lambda)\mathbf{j} + (8 + 2\lambda)\mathbf{k}\).
The vector \(\overrightarrow{AP}\) is:
\(\overrightarrow{AP} = (8 + 3\lambda)\mathbf{i} + (-3 - \lambda)\mathbf{j} + (4 + 2\lambda)\mathbf{k}\).
For \(\overrightarrow{AP}\) to be perpendicular to the direction vector of l, \(3\mathbf{i} - \mathbf{j} + 2\mathbf{k}\), their dot product must be zero:
\((8 + 3\lambda) \cdot 3 + (-3 - \lambda) \cdot (-1) + (4 + 2\lambda) \cdot 2 = 0\).
Simplifying, we get:
\(24 + 9\lambda + 3 + \lambda + 8 + 4\lambda = 0\).
\(14\lambda + 35 = 0\).
\(\lambda = -\frac{5}{2}\).
Substitute \(\lambda = -\frac{5}{2}\) back into \(\overrightarrow{OP}\) to find the foot of the perpendicular:
\(\overrightarrow{OP} = \left(9 + 3\left(-\frac{5}{2}\right)\right)\mathbf{i} + \left(-1 - \left(-\frac{5}{2}\right)\right)\mathbf{j} + \left(8 + 2\left(-\frac{5}{2}\right)\right)\mathbf{k}\).
\(\overrightarrow{OP} = \frac{3}{2}\mathbf{i} + \frac{3}{2}\mathbf{j} + 3\mathbf{k}\).
The reflection of A in l is found by using the formula:
\(\overrightarrow{OR} = 2\overrightarrow{OP} - \overrightarrow{OA}\).
\(\overrightarrow{OR} = 2\left(\frac{3}{2}\mathbf{i} + \frac{3}{2}\mathbf{j} + 3\mathbf{k}\right) - (\mathbf{i} + 2\mathbf{j} + 4\mathbf{k})\).
\(\overrightarrow{OR} = (3\mathbf{i} + 3\mathbf{j} + 6\mathbf{k}) - (\mathbf{i} + 2\mathbf{j} + 4\mathbf{k})\).
\(\overrightarrow{OR} = 2\mathbf{i} + \mathbf{j} + 2\mathbf{k}\).
