If a line passes through the points `(6,-7,-1)` and `(2,-3,1)`. Then what are the direction rations of the line

Options

A. `(4,-4,-2)`
B. `(4,4,2)`
C. `(-4,4,4)`
D. `(2,1,1)`

Answer

To find the direction ratios of the line that passes through the points \( A(6, -7, -1) \) and \( B(2, -3, 1) \), we can follow these steps:

### Step 1: Identify the coordinates of the points
We have two points:
- Point A: \( (x_1, y_1, z_1) = (6, -7, -1) \)
- Point B: \( (x_2, y_2, z_2) = (2, -3, 1) \)

### Step 2: Calculate the differences in coordinates
To find the direction ratios, we need to calculate the differences in the x, y, and z coordinates of the two points:
- \( \Delta x = x_2 - x_1 = 2 - 6 = -4 \)
- \( \Delta y = y_2 - y_1 = -3 - (-7) = -3 + 7 = 4 \)
- \( \Delta z = z_2 - z_1 = 1 - (-1) = 1 + 1 = 2 \)

### Step 3: Write the direction ratios
The direction ratios of the line can be represented as \( ( \Delta x, \Delta y, \Delta z ) \):
- Direction ratios = \( (-4, 4, 2) \)

### Step 4: Simplify the direction ratios (if necessary)
Direction ratios can be simplified by dividing by a common factor. In this case, we can divide by 2:
- Simplified direction ratios = \( (-2, 2, 1) \)

### Final Answer
The direction ratios of the line passing through the points \( (6, -7, -1) \) and \( (2, -3, 1) \) are \( (-4, 4, 2) \) or simplified as \( (-2, 2, 1) \).

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