If `a=hat(i)+hat(j)-2hat(k), b=2hat(i)-hat(j)+hat(k)` and `c=3hat(i)-hat(k)` and `c=ma+nb`, then `m+n` is equal to

Options

A. 0
B. 1
C. 2
D. `-1`

Explanation

To solve the problem, we start with the given vectors:

\[ \mathbf{a} = \hat{i} + \hat{j} - 2\hat{k} \]
\[ \mathbf{b} = 2\hat{i} - \hat{j} + \hat{k} \]
\[ \mathbf{c} = 3\hat{i} - \hat{k} \]

We are given that \(\mathbf{c} = m\mathbf{a} + n\mathbf{b}\). Our goal is to find the values of \(m\) and \(n\) and then compute \(m+n\).

### Step 1: Express \(m\mathbf{a} + n\mathbf{b}\)

First, we express \(m\mathbf{a}\) and \(n\mathbf{b}\):

\[ m\mathbf{a} = m(\hat{i} + \hat{j} - 2\hat{k}) = m\hat{i} + m\hat{j} - 2m\hat{k} \]
\[ n\mathbf{b} = n(2\hat{i} - \hat{j} + \hat{k}) = 2n\hat{i} - n\hat{j} + n\hat{k} \]

Now, we can combine these two expressions:

\[ m\mathbf{a} + n\mathbf{b} = (m + 2n)\hat{i} + (m - n)\hat{j} + (-2m + n)\hat{k} \]

### Step 2: Set the coefficients equal to those in \(\mathbf{c}\)

Since \(\mathbf{c} = 3\hat{i} + 0\hat{j} - 1\hat{k}\), we can equate the coefficients from both sides:

1. For \(\hat{i}\):
\[ m + 2n = 3 \quad \text{(1)} \]

2. For \(\hat{j}\):
\[ m - n = 0 \quad \text{(2)} \]

3. For \(\hat{k}\):
\[ -2m + n = -1 \quad \text{(3)} \]

### Step 3: Solve the equations

From equation (2), we can express \(m\) in terms of \(n\):

\[ m = n \quad \text{(4)} \]

Substituting equation (4) into equation (1):

\[ n + 2n = 3 \implies 3n = 3 \implies n = 1 \]

Using equation (4) again to find \(m\):

\[ m = n = 1 \]

### Step 4: Verify with the third equation

Now, we can verify our values of \(m\) and \(n\) using equation (3):

\[ -2(1) + 1 = -2 + 1 = -1 \]

This holds true, so our values are correct.

### Step 5: Calculate \(m+n\)

Finally, we calculate \(m+n\):

\[ m+n = 1 + 1 = 2 \]

Thus, the final answer is:

\[ \boxed{2} \]