Let `vec(A)=2hat(i)+hat(j),B=3hat(j)-hat(k)` and `vec(C )=6hat(i)-2hat(k)`. Find the value of `vec(A)-2vec(B)+3vec(C )`.

Options

A. `20 hat(i)+ 5hat(j) + 4hat(k)`
B. `20 hat(i)- 5hat(j) - 4hat(k)`
C. `4 hat(i)+ 5hat(j) + 20hat(k)`
D. `5 hat(i)+ 4hat(j) + 10hat(k)`

Answer

To solve the problem, we need to find the value of the expression \( \vec{A} - 2\vec{B} + 3\vec{C} \) given the vectors:

\[ \vec{A} = 2\hat{i} + \hat{j} \]
\[ \vec{B} = 3\hat{j} - \hat{k} \]
\[ \vec{C} = 6\hat{i} - 2\hat{k} \]

### Step 1: Write down the expression
We start with the expression:

\[ \vec{A} - 2\vec{B} + 3\vec{C} \]

### Step 2: Substitute the vectors
Substituting the values of \(\vec{A}\), \(\vec{B}\), and \(\vec{C}\):

\[ = (2\hat{i} + \hat{j}) - 2(3\hat{j} - \hat{k}) + 3(6\hat{i} - 2\hat{k}) \]

### Step 3: Distribute the coefficients
Now, we distribute the coefficients in the expression:

\[ = 2\hat{i} + \hat{j} - (6\hat{j} - 2\hat{k}) + (18\hat{i} - 6\hat{k}) \]

### Step 4: Combine like terms
Now we will combine the like terms for \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\):

1. **For \(\hat{i}\)**:
\[ 2\hat{i} + 18\hat{i} = 20\hat{i} \]

2. **For \(\hat{j}\)**:
\[ \hat{j} - 6\hat{j} = -5\hat{j} \]

3. **For \(\hat{k}\)**:
\[ -2\hat{k} - 6\hat{k} = -8\hat{k} \]

### Step 5: Write the final result
Putting it all together, we have:

\[ \vec{A} - 2\vec{B} + 3\vec{C} = 20\hat{i} - 5\hat{j} - 8\hat{k} \]

### Final Answer
Thus, the final result is:

\[ \vec{A} - 2\vec{B} + 3\vec{C} = 20\hat{i} - 5\hat{j} - 8\hat{k} \]

---