Given `veca+vecb=2hati," if "vecb=3hatj-hatk` then find out vector `veca` :-

Options

A. `2hati+3hatj+hatk`
B. `2hati-3hatj+hatk`
C. `hatj+hatk`
D. `2hati-hatj-hatk`

Answer

To find the vector \(\vec{a}\) given the equation \(\vec{a} + \vec{b} = 2\hat{i}\) and \(\vec{b} = 3\hat{j} - \hat{k}\), we can follow these steps:

### Step 1: Write down the given equations
We have:
\[ \vec{a} + \vec{b} = 2\hat{i} \]
and
\[ \vec{b} = 3\hat{j} - \hat{k} \]

### Step 2: Substitute \(\vec{b}\) into the equation
Substituting the expression for \(\vec{b}\) into the first equation:
\[ \vec{a} + (3\hat{j} - \hat{k}) = 2\hat{i} \]

### Step 3: Rearrange the equation to isolate \(\vec{a}\)
Now, we can rearrange the equation to solve for \(\vec{a}\):
\[ \vec{a} = 2\hat{i} - (3\hat{j} - \hat{k}) \]

### Step 4: Simplify the right-hand side
Distributing the negative sign:
\[ \vec{a} = 2\hat{i} - 3\hat{j} + \hat{k} \]

### Step 5: Write the final expression for \(\vec{a}\)
Thus, the vector \(\vec{a}\) is:
\[ \vec{a} = 2\hat{i} - 3\hat{j} + \hat{k} \]

### Final Answer
\(\vec{a} = 2\hat{i} - 3\hat{j} + \hat{k}\)

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