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 line is represented in symmetric form as \[ \frac{x+2}{5} = \frac{y+1}{2} = \frac{z-4}{-3} \]. Introducing a parameter \( t \), the parametric equations of the line are \( x = 5t - 2 \), \( y = 2t - 1 \), and \( z = -3t + 4 \). To find the foot of the perpendicular from the point \( (1, 1, 4) \) to this line, we minimize the distance between \( (1, 1, 4) \) and a general point \( (5t - 2, 2t - 1, -3t + 4) \) on the line. The distance formula is \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]. We find the value of \( t \) that minimizes this distance. Once \( t \) is determined, the coordinates of the foot of the perpendicular can be calculated.