Find the foot of the perpendicular from the point (1,2,3) to the line joining the points (6,7,7) and (9,9,5) .

Answer

To find the foot of the perpendicular from the point \( P(1, 2, 3) \) to the line joining the points \( B(6, 7, 7) \) and \( C(9, 9, 5) \), we can follow these steps:

### Step 1: Find the direction ratios of the line BC
The direction ratios of the line joining points \( B(6, 7, 7) \) and \( C(9, 9, 5) \) can be calculated as follows:

\[ \text{Direction ratios} = (9 - 6, 9 - 7, 5 - 7) = (3, 2, -2) \]

### Step 2: Write the parametric equations of the line BC
Using the direction ratios, we can write the parametric equations of the line \( BC \):

\[ x = 6 + 3r \]
\[ y = 7 + 2r \]
\[ z = 7 - 2r \]

### Step 3: Set up the point A on the line BC
Let the coordinates of point \( A \) (the foot of the perpendicular) be \( (x, y, z) \). From the parametric equations, we have:

\[ A(x, y, z) = (6 + 3r, 7 + 2r, 7 - 2r) \]

### Step 4: Find the direction ratios of line PA
The direction ratios of the line \( PA \) from point \( P(1, 2, 3) \) to point \( A(x, y, z) \) are:

\[ \text{Direction ratios of PA} = (x - 1, y - 2, z - 3) \]

Substituting the values of \( x, y, z \):

\[ = ((6 + 3r) - 1, (7 + 2r) - 2, (7 - 2r) - 3) = (5 + 3r, 5 + 2r, 4 - 2r) \]

### Step 5: Use the dot product to find the condition for perpendicularity
For the lines \( PA \) and \( BC \) to be perpendicular, their direction ratios must satisfy the condition:

\[ (5 + 3r) \cdot 3 + (5 + 2r) \cdot 2 + (4 - 2r) \cdot (-2) = 0 \]

Expanding this, we get:

\[ 3(5 + 3r) + 2(5 + 2r) - 2(4 - 2r) = 0 \]

### Step 6: Simplify the equation
Expanding the equation gives:

\[ 15 + 9r + 10 + 4r - 8 + 4r = 0 \]

Combining like terms:

\[ (9r + 4r + 4r) + (15 + 10 - 8) = 0 \]
\[ 17r + 17 = 0 \]

### Step 7: Solve for r
Solving for \( r \):

\[ 17r = -17 \implies r = -1 \]

### Step 8: Substitute r back to find the coordinates of A
Substituting \( r = -1 \) back into the parametric equations:

\[ x = 6 + 3(-1) = 6 - 3 = 3 \]
\[ y = 7 + 2(-1) = 7 - 2 = 5 \]
\[ z = 7 - 2(-1) = 7 + 2 = 9 \]

Thus, the coordinates of point \( A \) (the foot of the perpendicular) are \( (3, 5, 9) \).

### Final Answer
The foot of the perpendicular from the point \( (1, 2, 3) \) to the line joining the points \( (6, 7, 7) \) and \( (9, 9, 5) \) is \( (3, 5, 9) \).