The vector with initial point `P(2,-3,5)` and terminal point `Q(3,-4,7)` is :

Options

A. `hat(i)-hat(j)+2hat(k)`
B. `5hat(i)-7hat(j)+12hat(k)`
C. `-hat(i)+hat(j)-2hat(k)`
D. None of these

Answer

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

### Step 1: Identify the coordinates of points P and Q
- The coordinates of point \( P \) are \( (2, -3, 5) \).
- The coordinates of point \( Q \) are \( (3, -4, 7) \).

### Step 2: Represent the points in vector form
- The vector representation of point \( P \) is:
\[ \vec{P} = 2\hat{i} - 3\hat{j} + 5\hat{k} \]
- The vector representation of point \( Q \) is:
\[ \vec{Q} = 3\hat{i} - 4\hat{j} + 7\hat{k} \]

### Step 3: Find the vector \( \vec{PQ} \)
The vector \( \vec{PQ} \) can be found using the formula:
\[ \vec{PQ} = \vec{Q} - \vec{P} \]

### Step 4: Substitute the vector representations
Substituting the vector representations of \( P \) and \( Q \):
\[ \vec{PQ} = (3\hat{i} - 4\hat{j} + 7\hat{k}) - (2\hat{i} - 3\hat{j} + 5\hat{k}) \]

### Step 5: Perform the subtraction
Now, we subtract the corresponding components:
- For the \( \hat{i} \) component:
\[ 3 - 2 = 1\hat{i} \]
- For the \( \hat{j} \) component:
\[ -4 - (-3) = -4 + 3 = -1\hat{j} \]
- For the \( \hat{k} \) component:
\[ 7 - 5 = 2\hat{k} \]

### Step 6: Combine the results
Combining all the components, we get:
\[ \vec{PQ} = 1\hat{i} - 1\hat{j} + 2\hat{k} \]

### Final Result
Thus, the vector with initial point \( P \) and terminal point \( Q \) is:
\[ \vec{PQ} = \hat{i} - \hat{j} + 2\hat{k} \]

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