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, we will follow these steps:

### Step 1: Identify the coordinates of points A and B
The coordinates of point A are given as \( A(0, 4, -2) \) and the coordinates of point B are given as \( B(-2, 8, -4) \).

### Step 2: Write the position vectors for points A and B
The position vector for point A can be expressed as:
\[ \vec{A} = 0 \hat{i} + 4 \hat{j} - 2 \hat{k} = 0 \hat{i} + 4 \hat{j} - 2 \hat{k} \]
The position vector for point B can be expressed as:
\[ \vec{B} = -2 \hat{i} + 8 \hat{j} - 4 \hat{k} \]

### Step 3: Calculate the displacement vector from A to B
The displacement vector \( \vec{AB} \) from point A to point B is given by:
\[ \vec{AB} = \vec{B} - \vec{A} \]
Substituting the position vectors:
\[ \vec{AB} = (-2 \hat{i} + 8 \hat{j} - 4 \hat{k}) - (0 \hat{i} + 4 \hat{j} - 2 \hat{k}) \]

### Step 4: Perform the vector subtraction
Now, we will subtract the corresponding components:
\[ \vec{AB} = (-2 - 0) \hat{i} + (8 - 4) \hat{j} + (-4 + 2) \hat{k} \]
This simplifies to:
\[ \vec{AB} = -2 \hat{i} + 4 \hat{j} - 2 \hat{k} \]

### Step 5: Write the final answer
Thus, the displacement vector from point A to point B is:
\[ \vec{AB} = -2 \hat{i} + 4 \hat{j} - 2 \hat{k} \]