Two vectors are given by `vecA = 3hati + hatj+ 3hatk` and `vecB 3hati +5hatj - 2hatk.` Find the third vector `vecC` if `vecA+ 3vecB-vecC=0`

Options

A. `12hati+14hatj + 12hatk`
B. `13hati +17hatj + 12hatk`
C. `12hati +16hatj -3hatk`
D. `15hati +13hatj+ 4hatk`

Explanation

To solve for the vector \(\vec{C}\) given the equation \(\vec{A} + 3\vec{B} - \vec{C} = 0\), we can follow these steps:

### Step 1: Write down the given vectors
We have:
\[ \vec{A} = 3\hat{i} + \hat{j} + 3\hat{k} \]
\[ \vec{B} = 3\hat{i} + 5\hat{j} - 2\hat{k} \]

### Step 2: Substitute the vectors into the equation
We need to substitute \(\vec{A}\) and \(\vec{B}\) into the equation \(\vec{A} + 3\vec{B} - \vec{C} = 0\):
\[ (3\hat{i} + \hat{j} + 3\hat{k}) + 3(3\hat{i} + 5\hat{j} - 2\hat{k}) - \vec{C} = 0 \]

### Step 3: Calculate \(3\vec{B}\)
Now, calculate \(3\vec{B}\):
\[ 3\vec{B} = 3(3\hat{i} + 5\hat{j} - 2\hat{k}) = 9\hat{i} + 15\hat{j} - 6\hat{k} \]

### Step 4: Combine \(\vec{A}\) and \(3\vec{B}\)
Now, add \(\vec{A}\) and \(3\vec{B}\):
\[ \vec{A} + 3\vec{B} = (3\hat{i} + \hat{j} + 3\hat{k}) + (9\hat{i} + 15\hat{j} - 6\hat{k}) \]
Combine the components:
\[ = (3 + 9)\hat{i} + (1 + 15)\hat{j} + (3 - 6)\hat{k} \]
\[ = 12\hat{i} + 16\hat{j} - 3\hat{k} \]

### Step 5: Substitute back into the equation
Now substitute back into the equation:
\[ 12\hat{i} + 16\hat{j} - 3\hat{k} - \vec{C} = 0 \]

### Step 6: Solve for \(\vec{C}\)
Rearranging gives:
\[ \vec{C} = 12\hat{i} + 16\hat{j} - 3\hat{k} \]

### Final Answer
Thus, the vector \(\vec{C}\) is:
\[ \vec{C} = 12\hat{i} + 16\hat{j} - 3\hat{k} \]

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