Given three vectors `veca=6hati-3hatj,vecb=2hati-6hatj and vecc=-2hati+21hatj` such that `vecalpha=veca+vecb+vecc`. Then the resolution of te vector `vecalpha` into components with respect to `veca and vecb` is given by (A) `3veca-2vecb` (B) `2veca-3vecb` (C) `3vecb-2veca` (D) none of these

Options

A. `3veca - 2vecb`
B. `2veca - 3vecb`
C. `3vecb - 2veca`
D. None of these

Answer

To solve the problem, we need to find the vector \(\vec{\alpha}\) which is the sum of the given vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\). Then we will express \(\vec{\alpha}\) in terms of \(\vec{a}\) and \(\vec{b}\).

### Step 1: Calculate \(\vec{\alpha}\)

Given:
\[ \vec{a} = 6\hat{i} - 3\hat{j} \]
\[ \vec{b} = 2\hat{i} - 6\hat{j} \]
\[ \vec{c} = -2\hat{i} + 21\hat{j} \]

Now, we find \(\vec{\alpha}\):
\[ \vec{\alpha} = \vec{a} + \vec{b} + \vec{c} \]

Calculating the components:
- For \(\hat{i}\):
\[ 6 + 2 - 2 = 6 \]
- For \(\hat{j}\):
\[ -3 - 6 + 21 = 12 \]

Thus,
\[ \vec{\alpha} = 6\hat{i} + 12\hat{j} \]

### Step 2: Express \(\vec{\alpha}\) in terms of \(\vec{a}\) and \(\vec{b}\)

We want to express \(\vec{\alpha}\) as:
\[ \vec{\alpha} = x\vec{a} + y\vec{b} \]

Substituting the values of \(\vec{a}\) and \(\vec{b}\):
\[ 6\hat{i} + 12\hat{j} = x(6\hat{i} - 3\hat{j}) + y(2\hat{i} - 6\hat{j}) \]

Expanding the right-hand side:
\[ = (6x + 2y)\hat{i} + (-3x - 6y)\hat{j} \]

### Step 3: Set up equations for coefficients

Now we equate the coefficients of \(\hat{i}\) and \(\hat{j}\):
1. For \(\hat{i}\):
\[ 6 = 6x + 2y \quad \text{(1)} \]
2. For \(\hat{j}\):
\[ 12 = -3x - 6y \quad \text{(2)} \]

### Step 4: Solve the equations

From equation (1):
\[ 6 = 6x + 2y \implies 3 = 3x + y \implies y = 3 - 3x \quad \text{(3)} \]

Substituting equation (3) into equation (2):
\[ 12 = -3x - 6(3 - 3x) \]
\[ 12 = -3x - 18 + 18x \]
\[ 12 + 18 = 15x \]
\[ 30 = 15x \implies x = 2 \]

Now substituting \(x = 2\) back into equation (3):
\[ y = 3 - 3(2) = 3 - 6 = -3 \]

### Step 5: Write \(\vec{\alpha}\) in terms of \(\vec{a}\) and \(\vec{b}\)

Now we have \(x = 2\) and \(y = -3\):
\[ \vec{\alpha} = 2\vec{a} - 3\vec{b} \]

### Final Result

Thus, the resolution of the vector \(\vec{\alpha}\) into components with respect to \(\vec{a}\) and \(\vec{b}\) is:
\[ \vec{\alpha} = 2\vec{a} - 3\vec{b} \]

### Conclusion

The correct option is (B) \(2\vec{a} - 3\vec{b}\).