The length of the perpendicular drawn from the point `(3, -1, 11)` to the line `(x)/(2)=(y-2)/(3)=(z-3)/(4)` is

Options

A. `sqrt(66)`
B. `sqrt(29)`
C. `sqrt(33)`
D. `sqrt(53)`

Answer

To find the length of the perpendicular drawn from the point \( P(3, -1, 11) \) to the line given by the symmetric equations \( \frac{x}{2} = \frac{y-2}{3} = \frac{z-3}{4} \), we can follow these steps:

### Step 1: Parametrize the line
The symmetric equations can be rewritten in parametric form. Let \( t \) be the parameter. Then we have:
- \( x = 2t \)
- \( y = 3t + 2 \)
- \( z = 4t + 3 \)

### Step 2: Find a point on the line
From the parametric equations, we can find a point on the line by substituting \( t = 0 \):
- When \( t = 0 \), the point on the line is \( Q(0, 2, 3) \).

### Step 3: Find the direction vector of the line
The direction vector \( \vec{d} \) of the line can be derived from the coefficients of \( t \) in the parametric equations:
- \( \vec{d} = (2, 3, 4) \).

### Step 4: Find the vector from point \( P \) to point \( Q \)
The vector \( \vec{PQ} \) from point \( P(3, -1, 11) \) to point \( Q(0, 2, 3) \) is given by:
\[ \vec{PQ} = Q - P = (0 - 3, 2 - (-1), 3 - 11) = (-3, 3, -8). \]

### Step 5: Use the formula for the length of the perpendicular
The length of the perpendicular \( d \) from point \( P \) to the line can be calculated using the formula:
\[ d = \frac{|\vec{PQ} \cdot (\vec{d} \times \vec{PQ})|}{|\vec{d}|}. \]

### Step 6: Calculate \( \vec{d} \times \vec{PQ} \)
To find \( \vec{d} \times \vec{PQ} \):
\[ \vec{d} = (2, 3, 4), \quad \vec{PQ} = (-3, 3, -8). \]
Using the determinant to calculate the cross product:
\[ \vec{d} \times \vec{PQ} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & 4 \\ -3 & 3 & -8 \end{vmatrix} = \hat{i}(3 \cdot -8 - 4 \cdot 3) - \hat{j}(2 \cdot -8 - 4 \cdot -3) + \hat{k}(2 \cdot 3 - 3 \cdot -3). \]
Calculating each component:
- \( \hat{i}(-24 - 12) = -36\hat{i} \)
- \( \hat{j}(-16 + 12) = -4\hat{j} \)
- \( \hat{k}(6 + 9) = 15\hat{k} \)

Thus,
\[ \vec{d} \times \vec{PQ} = (-36, -4, 15). \]

### Step 7: Calculate the magnitude of \( \vec{d} \times \vec{PQ} \)
\[ |\vec{d} \times \vec{PQ}| = \sqrt{(-36)^2 + (-4)^2 + (15)^2} = \sqrt{1296 + 16 + 225} = \sqrt{1537}. \]

### Step 8: Calculate the magnitude of \( \vec{d} \)
\[ |\vec{d}| = \sqrt{2^2 + 3^2 + 4^2} = \sqrt{4 + 9 + 16} = \sqrt{29}. \]

### Step 9: Calculate the length of the perpendicular
Now substituting back into the formula for \( d \):
\[ d = \frac{|\vec{PQ} \cdot (\vec{d} \times \vec{PQ})|}{|\vec{d}|}. \]
First, we need to calculate \( \vec{PQ} \cdot (\vec{d} \times \vec{PQ}) \):
\[ \vec{PQ} \cdot (-36, -4, 15) = (-3)(-36) + (3)(-4) + (-8)(15) = 108 - 12 - 120 = -24. \]
Thus,
\[ d = \frac{|-24|}{\sqrt{29}} = \frac{24}{\sqrt{29}}. \]

### Final Step: Simplifying the result
To express the length in a more standard form, we can rationalize the denominator:
\[ d = \frac{24\sqrt{29}}{29}. \]

### Conclusion
The length of the perpendicular drawn from the point \( (3, -1, 11) \) to the line is \( \frac{24\sqrt{29}}{29} \).

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