Let `veca=3hati-hatj, vecb=hati-2hatj, vec c=-hati+7hatj` and `vecp=veca+vecb+vec c`. Then `vecP` in terms of `veca` and `vecb` is

Options

A. `2veca+3vecb`
B. `-2veca-3vecb`
C. `-2veca+3vecb`
D. `2veca-3hatb`

Answer

To solve the problem, we need to find the vector \(\vec{P}\) in terms of \(\vec{a}\) and \(\vec{b}\).

Given:
\[ \vec{a} = 3\hat{i} - \hat{j} \]
\[ \vec{b} = \hat{i} - 2\hat{j} \]
\[ \vec{c} = -\hat{i} + 7\hat{j} \]

The vector \(\vec{P}\) is defined as:
\[ \vec{P} = \vec{a} + \vec{b} + \vec{c} \]

### Step 1: Calculate \(\vec{P}\)
Substituting the values of \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\):
\[ \vec{P} = (3\hat{i} - \hat{j}) + (\hat{i} - 2\hat{j}) + (-\hat{i} + 7\hat{j}) \]

### Step 2: Combine the \(\hat{i}\) components
Combining the \(\hat{i}\) components:
\[ 3\hat{i} + \hat{i} - \hat{i} = (3 + 1 - 1)\hat{i} = 3\hat{i} \]

### Step 3: Combine the \(\hat{j}\) components
Combining the \(\hat{j}\) components:
\[ -\hat{j} - 2\hat{j} + 7\hat{j} = (-1 - 2 + 7)\hat{j} = 4\hat{j} \]

### Step 4: Write \(\vec{P}\) in terms of \(\hat{i}\) and \(\hat{j}\)
Thus, we have:
\[ \vec{P} = 3\hat{i} + 4\hat{j} \]

### Step 5: Express \(\vec{P}\) in terms of \(\vec{a}\) and \(\vec{b}\)
Now, we need to express \(\vec{P}\) in terms of \(\vec{a}\) and \(\vec{b}\). We will check the options provided:

1. **Option 1:** \(2\vec{a} + 3\vec{b}\)
\[ 2\vec{a} = 2(3\hat{i} - \hat{j}) = 6\hat{i} - 2\hat{j} \]
\[ 3\vec{b} = 3(\hat{i} - 2\hat{j}) = 3\hat{i} - 6\hat{j} \]
\[ 2\vec{a} + 3\vec{b} = (6\hat{i} - 2\hat{j}) + (3\hat{i} - 6\hat{j}) = 9\hat{i} - 8\hat{j} \quad \text{(not equal to } \vec{P}\text{)} \]

2. **Option 2:** \(-2\vec{a} - 3\vec{b}\)
\[ -2\vec{a} = -2(3\hat{i} - \hat{j}) = -6\hat{i} + 2\hat{j} \]
\[ -3\vec{b} = -3(\hat{i} - 2\hat{j}) = -3\hat{i} + 6\hat{j} \]
\[ -2\vec{a} - 3\vec{b} = (-6\hat{i} + 2\hat{j}) + (-3\hat{i} + 6\hat{j}) = -9\hat{i} + 8\hat{j} \quad \text{(not equal to } \vec{P}\text{)} \]

3. **Option 3:** \(-2\vec{a} + 3\vec{b}\)
\[ -2\vec{a} = -6\hat{i} + 2\hat{j} \]
\[ 3\vec{b} = 3\hat{i} - 6\hat{j} \]
\[ -2\vec{a} + 3\vec{b} = (-6\hat{i} + 2\hat{j}) + (3\hat{i} - 6\hat{j}) = -3\hat{i} - 4\hat{j} \quad \text{(not equal to } \vec{P}\text{)} \]

4. **Option 4:** \(2\vec{a} - 3\vec{b}\)
\[ 2\vec{a} = 6\hat{i} - 2\hat{j} \]
\[ -3\vec{b} = -3\hat{i} + 6\hat{j} \]
\[ 2\vec{a} - 3\vec{b} = (6\hat{i} - 2\hat{j}) + (-3\hat{i} + 6\hat{j}) = 3\hat{i} + 4\hat{j} \quad \text{(this is equal to } \vec{P}\text{)} \]

Thus, the correct expression for \(\vec{P}\) in terms of \(\vec{a}\) and \(\vec{b}\) is:
\[ \vec{P} = 2\vec{a} - 3\vec{b} \]