The `(x, y, z)` coordinates of two points A and B are given respectively as `(0, 4, -2) and (-2, 8, -4)`. The displacement vector form A to B is

Options

A. `-2hati + 4hatj - 2hatk`
B. `2hati - 4hatj + 2hatk`
C. `2hati+4hatj-2hatk`
D. `-2hati-4hatj - 2hatk`

Explanation

To find the displacement vector from point A to point B given their coordinates, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Coordinates**:
- Point A has coordinates \( A(0, 4, -2) \).
- Point B has coordinates \( B(-2, 8, -4) \).

2. **Write the Position Vectors**:
- The position vector of point A, denoted as \( \vec{r_A} \), can be expressed in vector form as:
\[ \vec{r_A} = 0\hat{i} + 4\hat{j} - 2\hat{k} = 0\hat{i} + 4\hat{j} - 2\hat{k} \]
- The position vector of point B, denoted as \( \vec{r_B} \), can be expressed as:
\[ \vec{r_B} = -2\hat{i} + 8\hat{j} - 4\hat{k} \]

3. **Calculate the Displacement Vector**:
- The displacement vector \( \vec{d} \) from point A to point B is given by:
\[ \vec{d} = \vec{r_B} - \vec{r_A} \]
- Substituting the position vectors:
\[ \vec{d} = (-2\hat{i} + 8\hat{j} - 4\hat{k}) - (0\hat{i} + 4\hat{j} - 2\hat{k}) \]

4. **Perform the Subtraction**:
- Distributing the subtraction:
\[ \vec{d} = (-2\hat{i} + 8\hat{j} - 4\hat{k}) - 0\hat{i} - 4\hat{j} + 2\hat{k} \]
- Combine like terms:
- For \( \hat{i} \): \( -2\hat{i} \)
- For \( \hat{j} \): \( 8\hat{j} - 4\hat{j} = 4\hat{j} \)
- For \( \hat{k} \): \( -4\hat{k} + 2\hat{k} = -2\hat{k} \)

5. **Final Result**:
- Therefore, the displacement vector \( \vec{d} \) from point A to point B is:
\[ \vec{d} = -2\hat{i} + 4\hat{j} - 2\hat{k} \]

### Final Answer:
The displacement vector from A to B is \( \vec{d} = -2\hat{i} + 4\hat{j} - 2\hat{k} \).

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