To find the length of the perpendicular from the point \( P(2, -1, 4) \) to the given straight line defined by the equations \( \frac{x+3}{10} = \frac{y-2}{-7} = \frac{z}{1} \), we can follow these steps:
### Step 1: Parametrize the Line
The given line can be expressed in parametric form. Let \( r \) be the parameter. Then, we have:
\[
x = 10r - 3, \quad y = -7r + 2, \quad z = r
\]
### Step 2: Identify the Direction Ratios
From the parametric equations, we can identify the direction ratios of the line:
\[
\text{Direction ratios} = (10, -7, 1)
\]
### Step 3: Find a Point on the Line
To find a specific point on the line, we can substitute \( r = 0 \):
\[
x = 10(0) - 3 = -3, \quad y = -7(0) + 2 = 2, \quad z = 0
\]
Thus, the point on the line is \( M(-3, 2, 0) \).
### Step 4: Find the Vector from Point P to Point M
Now, we find the vector \( PM \) from point \( P(2, -1, 4) \) to point \( M(-3, 2, 0) \):
\[
PM = M - P = (-3 - 2, 2 - (-1), 0 - 4) = (-5, 3, -4)
\]
### Step 5: Find the Length of the Perpendicular
The length of the perpendicular from point \( P \) to the line can be calculated using the formula:
\[
\text{Length} = \frac{|PM \cdot \text{Direction vector}|}{|\text{Direction vector}|}
\]
Where \( PM \cdot \text{Direction vector} \) is the dot product of the vectors, and \( |\text{Direction vector}| \) is the magnitude of the direction vector.
### Step 6: Calculate the Dot Product
Calculate the dot product \( PM \cdot (10, -7, 1) \):
\[
PM \cdot (10, -7, 1) = (-5)(10) + (3)(-7) + (-4)(1) = -50 - 21 - 4 = -75
\]
### Step 7: Calculate the Magnitude of the Direction Vector
Calculate the magnitude of the direction vector:
\[
|\text{Direction vector}| = \sqrt{10^2 + (-7)^2 + 1^2} = \sqrt{100 + 49 + 1} = \sqrt{150} = 5\sqrt{6}
\]
### Step 8: Calculate the Length of the Perpendicular
Now, substitute the values into the length formula:
\[
\text{Length} = \frac{| -75 |}{5\sqrt{6}} = \frac{75}{5\sqrt{6}} = \frac{15}{\sqrt{6}} = \frac{15\sqrt{6}}{6} = \frac{5\sqrt{6}}{2}
\]
Thus, the length of the perpendicular from the point \( (2, -1, 4) \) to the line is:
\[
\frac{5\sqrt{6}}{2}
\]
