If `a=6hat(i)+7hat(j)+7hat(k), b=3hat(i)+2hat(j)-2hat(k), P(1, 2, 3)`
Q. The image of the point P in the line ` r=a+lambdab` is

Options

A. `(11, 12, 11)`
B. `(5, 2, -7)`
C. `(5, 8, 15)`
D. `(17, 16, 7)`

Answer

To find the image of the point \( P(1, 2, 3) \) in the line given by the equation \( \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} \), where \( \mathbf{a} = 6\hat{i} + 7\hat{j} + 7\hat{k} \) and \( \mathbf{b} = 3\hat{i} + 2\hat{j} - 2\hat{k} \), we will follow these steps:

### Step 1: Write the equation of the line
The equation of the line in vector form is given by:
\[ \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} \]
Substituting the values of \( \mathbf{a} \) and \( \mathbf{b} \):
\[ \mathbf{r} = (6\hat{i} + 7\hat{j} + 7\hat{k}) + \lambda (3\hat{i} + 2\hat{j} - 2\hat{k}) \]
This can be expanded as:
\[ \mathbf{r} = (6 + 3\lambda)\hat{i} + (7 + 2\lambda)\hat{j} + (7 - 2\lambda)\hat{k} \]

### Step 2: Find the coordinates of the foot of the perpendicular (point L)
Let the coordinates of point \( L \) be \( (6 + 3\lambda, 7 + 2\lambda, 7 - 2\lambda) \). The coordinates of point \( P \) are \( (1, 2, 3) \).

### Step 3: Find the direction ratios of line segment \( LP \)
The direction ratios of the line segment \( LP \) can be calculated as follows:
\[ \text{Direction ratios of } LP = (6 + 3\lambda - 1, 7 + 2\lambda - 2, 7 - 2\lambda - 3) \]
This simplifies to:
\[ (5 + 3\lambda, 5 + 2\lambda, 4 - 2\lambda) \]

### Step 4: Set up the condition for perpendicularity
Since \( LP \) is perpendicular to the line, the dot product of the direction ratios of \( LP \) and the direction ratios of \( \mathbf{b} \) must be zero:
\[ (5 + 3\lambda)\cdot 3 + (5 + 2\lambda)\cdot 2 + (4 - 2\lambda)\cdot (-2) = 0 \]

### Step 5: Expand and simplify the equation
Expanding the equation gives:
\[ 15 + 9\lambda + 10 + 4\lambda - 8 + 4\lambda = 0 \]
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 point L
Substituting \( \lambda = -1 \) into the coordinates of \( L \):
\[ L = (6 + 3(-1), 7 + 2(-1), 7 - 2(-1)) = (6 - 3, 7 - 2, 7 + 2) = (3, 5, 9) \]

### Step 8: Write the position vector of point L
The position vector of point \( L \) is:
\[ \mathbf{L} = 3\hat{i} + 5\hat{j} + 9\hat{k} \]

### Final Answer
The image of the point \( P \) in the line \( \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} \) is given by the position vector:
\[ \mathbf{L} = 3\hat{i} + 5\hat{j} + 9\hat{k} \]

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