The vector joining the points A (1,1,-1) and B (2,-3,4) & pointing A to B is

Options

A. `hati(-1) +4hat(j)-5hat(k)`
B. `hat(i)+4hat(j)+5hat(k)`
C. `hat(i)-4hat(j)+5hat(k)`
D. `hati(-1) -4hathat(j)-5hat(k)`

Answer

To find the vector joining the points A(1, 1, -1) and B(2, -3, 4) and pointing from A to B, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Coordinates of Points A and B**:
- Point A has coordinates \( A(1, 1, -1) \).
- Point B has coordinates \( B(2, -3, 4) \).

2. **Write the Position Vectors**:
- The position vector of point A, denoted as \( \vec{A} \), can be written as:
\[ \vec{A} = 1\hat{i} + 1\hat{j} - 1\hat{k} = \hat{i} + \hat{j} - \hat{k} \]
- The position vector of point B, denoted as \( \vec{B} \), can be written as:
\[ \vec{B} = 2\hat{i} - 3\hat{j} + 4\hat{k} \]

3. **Calculate the Vector from A to B**:
- The vector \( \vec{AB} \) pointing from A to B is given by:
\[ \vec{AB} = \vec{B} - \vec{A} \]
- Substituting the position vectors:
\[ \vec{AB} = (2\hat{i} - 3\hat{j} + 4\hat{k}) - (\hat{i} + \hat{j} - \hat{k}) \]

4. **Perform the Subtraction**:
- Distributing the negative sign:
\[ \vec{AB} = 2\hat{i} - 3\hat{j} + 4\hat{k} - \hat{i} - \hat{j} + \hat{k} \]
- Combine like terms:
- For \( \hat{i} \): \( 2 - 1 = 1 \)
- For \( \hat{j} \): \( -3 - 1 = -4 \)
- For \( \hat{k} \): \( 4 + 1 = 5 \)
- Therefore, the resulting vector is:
\[ \vec{AB} = 1\hat{i} - 4\hat{j} + 5\hat{k} \]

5. **Final Result**:
- The vector joining points A and B, pointing from A to B, is:
\[ \vec{AB} = \hat{i} - 4\hat{j} + 5\hat{k} \]