To find the length of the perpendicular from the point \( P(1, 6, 3) \) to the line given by the equations \( \frac{x}{1} = \frac{y-1}{2} = \frac{z-2}{3} \), we can follow these steps:
### Step 1: Parametrize the Line
The line can be expressed in parametric form. Let \( \lambda \) be the parameter. Then, we can write:
- \( x = \lambda \)
- \( y = 2\lambda + 1 \)
- \( z = 3\lambda + 2 \)
### Step 2: Find the Coordinates of Point Q on the Line
The coordinates of any point \( Q \) on the line can be represented as:
\[ Q(\lambda, 2\lambda + 1, 3\lambda + 2) \]
### Step 3: Find the Direction Ratios of PQ
The direction ratios of the line segment \( PQ \) (from point \( P(1, 6, 3) \) to point \( Q \)) can be calculated as:
- \( PQ_x = \lambda - 1 \)
- \( PQ_y = (2\lambda + 1) - 6 = 2\lambda - 5 \)
- \( PQ_z = (3\lambda + 2) - 3 = 3\lambda - 1 \)
Thus, the direction ratios of \( PQ \) are:
\[ (PQ_x, PQ_y, PQ_z) = (\lambda - 1, 2\lambda - 5, 3\lambda - 1) \]
### Step 4: Direction Ratios of the Line
The direction ratios of the line are given as \( (1, 2, 3) \).
### Step 5: Use the Perpendicular Condition
For \( PQ \) to be perpendicular to the line, the dot product of their direction ratios must be zero:
\[
(\lambda - 1) \cdot 1 + (2\lambda - 5) \cdot 2 + (3\lambda - 1) \cdot 3 = 0
\]
### Step 6: Expand and Simplify the Equation
Expanding the equation:
\[
\lambda - 1 + 4\lambda - 10 + 9\lambda - 3 = 0
\]
Combining like terms:
\[
(1 + 4 + 9)\lambda - (1 + 10 + 3) = 0
\]
\[
14\lambda - 14 = 0
\]
### Step 7: Solve for \( \lambda \)
Solving for \( \lambda \):
\[
14\lambda = 14 \implies \lambda = 1
\]
### Step 8: Find the Coordinates of Point Q
Substituting \( \lambda = 1 \) back into the parametric equations gives:
- \( x = 1 \)
- \( y = 2(1) + 1 = 3 \)
- \( z = 3(1) + 2 = 5 \)
Thus, the coordinates of point \( Q \) are \( (1, 3, 5) \).
### Step 9: Calculate the Length of the Perpendicular PQ
The length of the perpendicular \( PQ \) can be calculated using the distance formula:
\[
PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}
\]
Substituting the coordinates:
\[
PQ = \sqrt{(1 - 1)^2 + (3 - 6)^2 + (5 - 3)^2}
\]
Calculating each term:
\[
= \sqrt{0^2 + (-3)^2 + 2^2} = \sqrt{0 + 9 + 4} = \sqrt{13}
\]
### Final Answer
The length of the perpendicular from the point \( (1, 6, 3) \) to the line is \( \sqrt{13} \).
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