The vector equation of the line `(x-5)/(3) = ( y +4)/(7) = (z-6)/(2)` is `vecr = 5 hati - 4 hatj + 6 hatk + lamda( 3 hati + 7 hatj + 2hatk)`

Answer

To determine if the given vector equation of the line is correct, we need to analyze the provided Cartesian form of the line and convert it into vector form.

### Step-by-Step Solution:

1. **Identify the Cartesian Equation**: The given equation of the line is:
\[ \frac{x - 5}{3} = \frac{y + 4}{7} = \frac{z - 6}{2} \]

2. **Understand the Form**: The general form of the equation of a line in three-dimensional space can be expressed as:
\[ \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c} \]
where \((x_1, y_1, z_1)\) is a point on the line and \((a, b, c)\) are the direction ratios of the line.

3. **Extract Point and Direction Ratios**:
- From the equation, we can identify:
- \(x_1 = 5\)
- \(y_1 = -4\)
- \(z_1 = 6\)
- Direction ratios \(a = 3\), \(b = 7\), \(c = 2\).

4. **Write the Position Vector**: The position vector of the point \((5, -4, 6)\) is:
\[ \vec{r_0} = 5\hat{i} - 4\hat{j} + 6\hat{k} \]

5. **Write the Direction Vector**: The direction vector corresponding to the direction ratios is:
\[ \vec{d} = 3\hat{i} + 7\hat{j} + 2\hat{k} \]

6. **Formulate the Vector Equation**: The vector equation of the line can be written as:
\[ \vec{r} = \vec{r_0} + \lambda \vec{d} \]
Substituting the values, we get:
\[ \vec{r} = (5\hat{i} - 4\hat{j} + 6\hat{k}) + \lambda(3\hat{i} + 7\hat{j} + 2\hat{k}) \]

7. **Final Expression**: Therefore, the vector equation of the line is:
\[ \vec{r} = 5\hat{i} - 4\hat{j} + 6\hat{k} + \lambda(3\hat{i} + 7\hat{j} + 2\hat{k}) \]

### Conclusion:
The vector equation provided in the question is indeed correct.