The foot of the perpendicular from the point (1,2,3) on the line `vecr=(6hati+7hatj+7hatk)+lamda(3hati+2hatj-2hatk)` has the coordinates

Options

A. (5,8,15)
B. (8,5,15)
C. (3,5,9)
D. (3,5,-9)

Explanation

To find the foot of the perpendicular from the point \( P(1, 2, 3) \) to the line given by the vector equation \( \vec{r} = 6\hat{i} + 7\hat{j} + 7\hat{k} + \lambda(3\hat{i} + 2\hat{j} - 2\hat{k}) \), we will follow these steps:

### Step 1: Determine the coordinates of a point on the line
The line can be expressed in terms of \( \lambda \):
\[ \vec{Q} = (6 + 3\lambda, 7 + 2\lambda, 7 - 2\lambda) \]
This gives us the coordinates of point \( Q \) on the line in terms of \( \lambda \).

### Step 2: Find the vector \( \vec{PQ} \)
The vector \( \vec{PQ} \) from point \( P(1, 2, 3) \) to point \( Q \) is given by:
\[ \vec{PQ} = Q - P = (6 + 3\lambda - 1, 7 + 2\lambda - 2, 7 - 2\lambda - 3) \]
This simplifies to:
\[ \vec{PQ} = (5 + 3\lambda, 5 + 2\lambda, 4 - 2\lambda) \]

### Step 3: Find the direction ratios of the line
The direction ratios of the line are given by the coefficients of \( \lambda \) in the line equation:
\[ \vec{B} = (3, 2, -2) \]

### Step 4: Set up the dot product condition
Since \( \vec{PQ} \) is perpendicular to the line, their dot product must equal zero:
\[ \vec{PQ} \cdot \vec{B} = 0 \]
Calculating the dot product:
\[ (5 + 3\lambda) \cdot 3 + (5 + 2\lambda) \cdot 2 + (4 - 2\lambda) \cdot (-2) = 0 \]

### Step 5: Expand and simplify the equation
Expanding the dot product:
\[ 15 + 9\lambda + 10 + 4\lambda - 8 + 4\lambda = 0 \]
Combining like terms:
\[ (9\lambda + 4\lambda + 4\lambda) + (15 + 10 - 8) = 0 \]
This simplifies to:
\[ 17\lambda + 17 = 0 \]

### Step 6: Solve for \( \lambda \)
Solving for \( \lambda \):
\[ 17\lambda = -17 \implies \lambda = -1 \]

### Step 7: Substitute \( \lambda \) back to find coordinates of \( Q \)
Now substitute \( \lambda = -1 \) back into the coordinates of \( Q \):
\[ Q = (6 + 3(-1), 7 + 2(-1), 7 - 2(-1)) \]
This simplifies to:
\[ Q = (6 - 3, 7 - 2, 7 + 2) = (3, 5, 9) \]

### Conclusion
The coordinates of the foot of the perpendicular from the point \( (1, 2, 3) \) to the line are \( (3, 5, 9) \).

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