If the root of perpendicular from `A(3,1,0)` on a line passing through `B(1,alpha,7)` is `C(17//3,5//3,7//3)`, then `alpha`= _________

Answer

To find the value of \( \alpha \) in the given problem, we will follow these steps:

### Step 1: Identify the points and vectors
We have the following points:
- Point \( A(3, 1, 0) \)
- Point \( B(1, \alpha, 7) \)
- Foot of the perpendicular \( C\left(\frac{17}{3}, \frac{5}{3}, \frac{7}{3}\right) \)

We need to find the vectors \( \overrightarrow{AC} \) and \( \overrightarrow{BC} \).

### Step 2: Calculate the vector \( \overrightarrow{AC} \)
The vector \( \overrightarrow{AC} \) can be calculated as follows:
\[ \overrightarrow{AC} = C - A = \left(\frac{17}{3} - 3, \frac{5}{3} - 1, \frac{7}{3} - 0\right) \]
Calculating each component:
\[ \overrightarrow{AC} = \left(\frac{17}{3} - \frac{9}{3}, \frac{5}{3} - \frac{3}{3}, \frac{7}{3} - 0\right) = \left(\frac{8}{3}, \frac{2}{3}, \frac{7}{3}\right) \]

### Step 3: Calculate the vector \( \overrightarrow{BC} \)
The vector \( \overrightarrow{BC} \) can be calculated as follows:
\[ \overrightarrow{BC} = C - B = \left(\frac{17}{3} - 1, \frac{5}{3} - \alpha, \frac{7}{3} - 7\right) \]
Calculating each component:
\[ \overrightarrow{BC} = \left(\frac{17}{3} - \frac{3}{3}, \frac{5}{3} - \alpha, \frac{7}{3} - \frac{21}{3}\right) = \left(\frac{14}{3}, \frac{5}{3} - \alpha, -\frac{14}{3}\right) \]

### Step 4: Use the dot product to find \( \alpha \)
Since \( \overrightarrow{AC} \) and \( \overrightarrow{BC} \) are perpendicular, their dot product must equal zero:
\[ \overrightarrow{AC} \cdot \overrightarrow{BC} = 0 \]
Calculating the dot product:
\[ \left(\frac{8}{3}, \frac{2}{3}, \frac{7}{3}\right) \cdot \left(\frac{14}{3}, \frac{5}{3} - \alpha, -\frac{14}{3}\right) = 0 \]
Calculating each term:
\[ \frac{8}{3} \cdot \frac{14}{3} + \frac{2}{3} \left(\frac{5}{3} - \alpha\right) + \frac{7}{3} \cdot \left(-\frac{14}{3}\right) = 0 \]
This simplifies to:
\[ \frac{112}{9} + \frac{2}{3} \left(\frac{5}{3} - \alpha\right) - \frac{98}{9} = 0 \]
Combining terms:
\[ \frac{112}{9} - \frac{98}{9} + \frac{10}{9} - \frac{2\alpha}{3} = 0 \]
\[ \frac{24}{9} + \frac{10}{9} - \frac{2\alpha}{3} = 0 \]
\[ \frac{34}{9} - \frac{2\alpha}{3} = 0 \]

### Step 5: Solve for \( \alpha \)
Now, we can solve for \( \alpha \):
\[ \frac{2\alpha}{3} = \frac{34}{9} \]
Multiplying both sides by 3:
\[ 2\alpha = \frac{34}{3} \]
Dividing both sides by 2:
\[ \alpha = \frac{34}{6} = \frac{17}{3} \]

### Final Answer
Thus, the value of \( \alpha \) is \( 4 \).