Let the coordinates of the points A, B, C be (1,8,4), (0,-11,4) and (2,-3,1) respectively. What are the coordinates of the point D which is the foot of the perpendicular from A on BC?

Options

A. `(3,4,-2)`
B. `(4,-2,5)`
C. `(4,5,-2)`
D. `(2,4,5)`

Answer

To find the coordinates of point D, which is the foot of the perpendicular from point A to line BC, we can follow these steps:

### Step 1: Identify the Coordinates
The coordinates of the points are:
- A(1, 8, 4)
- B(0, -11, 4)
- C(2, -3, 1)

### Step 2: Find the Direction Ratios of Line BC
To find the direction ratios of line BC, we can use the coordinates of points B and C.

The direction ratios (dr) of line BC can be calculated as:
- \( \text{dr}_{BC} = (x_C - x_B, y_C - y_B, z_C - z_B) \)
- \( \text{dr}_{BC} = (2 - 0, -3 - (-11), 1 - 4) \)
- \( \text{dr}_{BC} = (2, 8, -3) \)

### Step 3: Parametric Equations of Line BC
Using the point B and the direction ratios, we can write the parametric equations for line BC:
- \( x = 0 + 2t = 2t \)
- \( y = -11 + 8t \)
- \( z = 4 - 3t \)

### Step 4: General Point on Line BC
Let the general point D on line BC be represented as:
- \( D(2t, -11 + 8t, 4 - 3t) \)

### Step 5: Find the Direction Ratios of Line AD
The direction ratios of line AD can be calculated using the coordinates of points A and D:
- \( \text{dr}_{AD} = (x_D - x_A, y_D - y_A, z_D - z_A) \)
- \( \text{dr}_{AD} = (2t - 1, (-11 + 8t) - 8, (4 - 3t) - 4) \)
- \( \text{dr}_{AD} = (2t - 1, 8t - 19, -3t) \)

### Step 6: Perpendicular Condition
Since AD is perpendicular to BC, the dot product of their direction ratios must equal zero:
- \( \text{dr}_{AD} \cdot \text{dr}_{BC} = 0 \)
- \( (2, 8, -3) \cdot (2t - 1, 8t - 19, -3t) = 0 \)

Calculating the dot product:
- \( 2(2t - 1) + 8(8t - 19) - 3(-3t) = 0 \)
- \( 4t - 2 + 64t - 152 + 9t = 0 \)
- \( 77t - 154 = 0 \)

### Step 7: Solve for t
Now, solve for t:
- \( 77t = 154 \)
- \( t = 2 \)

### Step 8: Find Coordinates of Point D
Substituting \( t = 2 \) back into the parametric equations for D:
- \( x_D = 2(2) = 4 \)
- \( y_D = -11 + 8(2) = -11 + 16 = 5 \)
- \( z_D = 4 - 3(2) = 4 - 6 = -2 \)

Thus, the coordinates of point D are:
- \( D(4, 5, -2) \)

### Final Answer
The coordinates of point D are \( (4, 5, -2) \).

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