The vector having initial and teminal points ( 2,5,0) and ( - 3,7,4) respectively is

Options

A. `- 5 hat(i) + 2 hat(j) + 4 hat(k)`
B. `5 hat(i)- 2 hat(j) - 4 hat(k)`
C. `5 hat(i) + 2 hat(j) + 4 hat(k)`
D. `- 5 hat(i) + 2 hat(i) - 4hat(k)`

Answer

To find the vector having initial and terminal points (2, 5, 0) and (-3, 7, 4) respectively, we can follow these steps:

### Step 1: Identify the Initial and Terminal Points
Let the initial point \( A \) be \( (2, 5, 0) \) and the terminal point \( B \) be \( (-3, 7, 4) \).

### Step 2: Write the Position Vectors
The position vector of point \( A \) (initial point) can be written as:
\[ \vec{OA} = 2\hat{i} + 5\hat{j} + 0\hat{k} \]
The position vector of point \( B \) (terminal point) can be written as:
\[ \vec{OB} = -3\hat{i} + 7\hat{j} + 4\hat{k} \]

### Step 3: Find the Vector \( \vec{AB} \)
The vector \( \vec{AB} \) can be found using the formula:
\[ \vec{AB} = \vec{OB} - \vec{OA} \]
Substituting the position vectors:
\[ \vec{AB} = (-3\hat{i} + 7\hat{j} + 4\hat{k}) - (2\hat{i} + 5\hat{j} + 0\hat{k}) \]

### Step 4: Perform the Subtraction
Now, we perform the subtraction component-wise:
\[ \vec{AB} = (-3 - 2)\hat{i} + (7 - 5)\hat{j} + (4 - 0)\hat{k} \]
This simplifies to:
\[ \vec{AB} = -5\hat{i} + 2\hat{j} + 4\hat{k} \]

### Step 5: Write the Final Answer
Thus, the vector having initial point \( (2, 5, 0) \) and terminal point \( (-3, 7, 4) \) is:
\[ \vec{AB} = -5\hat{i} + 2\hat{j} + 4\hat{k} \]