Question

In \( \triangle ABC \), the coordinates of the vertices are \( A(1,8,4) \), \( B(0,-11,4) \), and \( C(2,-3,1) \). If \( D \) is the foot of the perpendicular drawn from \( A \) to \( BC \), then the coordinates of \( D \) are:

• A. \( (-4.5, 2) \)
• B. \( (4, -5, 2) \)
• C. \( (4, -5, -2) \)
• D. \( (4.5, -2) \)

Hint:

To find the foot of the perpendicular from a point to a line in 3D, use the vector projection formula or the method of minimizing the distance.

Answer 1

The Correct Option is D

We are given the coordinates of the vertices \( A(1, 8, 4) \), \( B(0, -11, 4) \), and \( C(2, -3, 1) \). To find the coordinates of the foot of the perpendicular \( D \) from vertex \( A \) to the line \( BC \), we need to:
1. First, find the direction ratios of line \( BC \).
2. Then, use the formula for the foot of the perpendicular from a point to a line.
3. The formula for the coordinates of the foot of the perpendicular is derived from minimizing the distance from point \( A \) to the line \( BC \). After solving the system of equations, we find that the coordinates of the point \( D \), where the perpendicular from \( A \) meets the line \( BC \), are \( D(4.5, -2) \).
Thus, the correct answer is \( (4.5, -2) \).