Question

Find the foot of the perpendicular drawn from the point \( (1, 1, 4) \) on the line \( \frac{x+2}{5} = \frac{y+1}{2} = \frac{z-4}{-3} \).

Hint:

To find the foot of the perpendicular from a point to a line, parametrize the line and minimize the distance between the point and any point on the line using the distance formula.

Answer 1

The equation of the line is given in symmetric form: \[ \frac{x+2}{5} = \frac{y+1}{2} = \frac{z-4}{-3} \] Let the common parameter be \( t \). Then, parametrize the coordinates of a point on the line as: \[ x = 5t - 2, \quad y = 2t - 1, \quad z = -3t + 4 \] Now, the distance between the point \( (1, 1, 4) \) and a point on the line \( (5t - 2, 2t - 1, -3t + 4) \) must be minimized to find the foot of the perpendicular. Using the formula for distance between two points: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \] Minimizing this distance, we solve for \( t \) that gives the minimum distance. After solving, the coordinates of the foot of the perpendicular are found.