The vector equation of the line passing through the points `A(3,4,-7) and B (1,-1,6)` is

Options

A. `overset(to) ( r) = 3 hat(i) + 4 hat(j) - 7 hat(k) + lambda(hat (i) - hat(j) +6 hat(k) )`
B. `overset(to) ( r) = 3 hat(i) + 4 hat(j) -7 hat(k) + lambda ( -2 hat(i) -5 hat(j) + 13 hat(k) )`
C. `overset(to) ( r) =hat(i) - hat(j) + 6 hat(k) + lambda (3 hat(i) + 4 hat(j) - 7 hat(k) )`
D. `overset(to) (r ) = hat(i) - hat(j) + 6 hat( k ) + lambda (4 hat(i) + 3hat(j) - hat(k) )`

Answer

To find the vector equation of the line passing through the points \( A(3, 4, -7) \) and \( B(1, -1, 6) \), we can follow these steps:

### Step 1: Identify the points
We have two points:
- Point \( A \) with coordinates \( (3, 4, -7) \)
- Point \( B \) with coordinates \( (1, -1, 6) \)

### Step 2: Write the position vector of point A
The position vector \( \vec{A} \) corresponding to point \( A \) can be expressed as:
\[ \vec{A} = 3\hat{i} + 4\hat{j} - 7\hat{k} \]

### Step 3: Find the direction vector
The direction vector \( \vec{AB} \) from point \( A \) to point \( B \) can be calculated using the formula:
\[ \vec{AB} = \vec{B} - \vec{A} \]
First, we find the coordinates of point \( B \) in vector form:
\[ \vec{B} = 1\hat{i} - 1\hat{j} + 6\hat{k} \]
Now, we can calculate \( \vec{AB} \):
\[ \vec{AB} = (1 - 3)\hat{i} + (-1 - 4)\hat{j} + (6 + 7)\hat{k} \]
This simplifies to:
\[ \vec{AB} = -2\hat{i} - 5\hat{j} + 13\hat{k} \]

### Step 4: Write the vector equation of the line
The vector equation of a line can be expressed as:
\[ \vec{r} = \vec{A} + \lambda \vec{AB} \]
Substituting the values we have:
\[ \vec{r} = (3\hat{i} + 4\hat{j} - 7\hat{k}) + \lambda (-2\hat{i} - 5\hat{j} + 13\hat{k}) \]

### Step 5: Finalize the equation
Thus, the vector equation of the line is:
\[ \vec{r} = (3 - 2\lambda)\hat{i} + (4 - 5\lambda)\hat{j} + (-7 + 13\lambda)\hat{k} \]

### Conclusion
The vector equation of the line passing through points \( A(3, 4, -7) \) and \( B(1, -1, 6) \) is:
\[ \vec{r} = (3 - 2\lambda)\hat{i} + (4 - 5\lambda)\hat{j} + (-7 + 13\lambda)\hat{k} \]