To find the perpendicular distance from point C to the line segment AB, we can follow these steps:
### Step 1: Determine the direction ratios of line AB
Given points A(1, 1, 1) and B(3, 7, 4), we can find the direction ratios of line AB. The direction ratios can be calculated as follows:
\[
\text{Direction ratios of } AB = (B_x - A_x, B_y - A_y, B_z - A_z) = (3 - 1, 7 - 1, 4 - 1) = (2, 6, 3)
\]
### Step 2: Write the parametric equations of line AB
Using point A and the direction ratios, we can write the parametric equations for the line AB:
\[
x = 1 + 2\lambda, \quad y = 1 + 6\lambda, \quad z = 1 + 3\lambda
\]
### Step 3: Determine a point Q on line AB
Let Q be a point on line AB represented by the parameter \(\lambda\):
\[
Q(\lambda) = (1 + 2\lambda, 1 + 6\lambda, 1 + 3\lambda)
\]
### Step 4: Calculate the direction ratios of line QC
The direction ratios of line QC from point C(-1, 3, 0) to point Q are:
\[
\text{Direction ratios of } QC = (Q_x - C_x, Q_y - C_y, Q_z - C_z) = ((1 + 2\lambda) - (-1), (1 + 6\lambda) - 3, (1 + 3\lambda) - 0)
\]
\[
= (2\lambda + 2, 6\lambda - 2, 3\lambda + 1)
\]
### Step 5: Set up the condition for perpendicularity
For line AB and line QC to be perpendicular, the dot product of their direction ratios must equal zero:
\[
(2, 6, 3) \cdot (2\lambda + 2, 6\lambda - 2, 3\lambda + 1) = 0
\]
Calculating the dot product:
\[
2(2\lambda + 2) + 6(6\lambda - 2) + 3(3\lambda + 1) = 0
\]
\[
4\lambda + 4 + 36\lambda - 12 + 9\lambda + 3 = 0
\]
\[
(4 + 36 + 9)\lambda + (4 - 12 + 3) = 0
\]
\[
49\lambda - 5 = 0
\]
\[
49\lambda = 5 \quad \Rightarrow \quad \lambda = \frac{5}{49}
\]
### Step 6: Find the coordinates of point Q
Substituting \(\lambda = \frac{5}{49}\) back into the parametric equations for Q:
\[
Q = \left(1 + 2\left(\frac{5}{49}\right), 1 + 6\left(\frac{5}{49}\right), 1 + 3\left(\frac{5}{49}\right)\right)
\]
\[
= \left(1 + \frac{10}{49}, 1 + \frac{30}{49}, 1 + \frac{15}{49}\right)
\]
\[
= \left(\frac{59}{49}, \frac{79}{49}, \frac{64}{49}\right)
\]
### Step 7: Calculate the distance QC
Now, we calculate the distance from point C to point Q using the distance formula:
\[
d = \sqrt{(Q_x - C_x)^2 + (Q_y - C_y)^2 + (Q_z - C_z)^2}
\]
\[
= \sqrt{\left(\frac{59}{49} - (-1)\right)^2 + \left(\frac{79}{49} - 3\right)^2 + \left(\frac{64}{49} - 0\right)^2}
\]
\[
= \sqrt{\left(\frac{59}{49} + \frac{49}{49}\right)^2 + \left(\frac{79}{49} - \frac{147}{49}\right)^2 + \left(\frac{64}{49}\right)^2}
\]
\[
= \sqrt{\left(\frac{108}{49}\right)^2 + \left(-\frac{68}{49}\right)^2 + \left(\frac{64}{49}\right)^2}
\]
\[
= \sqrt{\frac{11664}{2401} + \frac{4624}{2401} + \frac{4096}{2401}}
\]
\[
= \sqrt{\frac{11664 + 4624 + 4096}{2401}} = \sqrt{\frac{20484}{2401}} = \frac{\sqrt{20484}}{49}
\]
### Final Answer
Thus, the perpendicular distance from point C to line AB is:
\[
\frac{\sqrt{20484}}{49} \text{ units}
\]
