Find the vector having initial and terminal point as (2, 5,0) and (-3, 7, 4).

Answer

To find the vector with initial point \( A(2, 5, 0) \) and terminal point \( B(-3, 7, 4) \), we can follow these steps:

### Step 1: Identify the position vectors
The position vector of point \( A \) is given by:
\[ \vec{OA} = 2\hat{i} + 5\hat{j} + 0\hat{k} \]
The position vector of point \( B \) is given by:
\[ \vec{OB} = -3\hat{i} + 7\hat{j} + 4\hat{k} \]

### Step 2: Use the formula for the vector from \( A \) to \( B \)
The vector \( \vec{AB} \) can be found using the formula:
\[ \vec{AB} = \vec{OB} - \vec{OA} \]

### Step 3: Substitute the position vectors
Substituting the position vectors into the formula:
\[ \vec{AB} = (-3\hat{i} + 7\hat{j} + 4\hat{k}) - (2\hat{i} + 5\hat{j} + 0\hat{k}) \]

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

### Step 5: Simplify the expression
This simplifies to:
\[ \vec{AB} = -5\hat{i} + 2\hat{j} + 4\hat{k} \]

### Final Answer
Thus, the vector from point \( A \) to point \( B \) is:
\[ \vec{AB} = -5\hat{i} + 2\hat{j} + 4\hat{k} \]

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