`veca` and `vecc` are unit collinear vectors and `|vecb|=6`, then `vecb-3vecc = lambda veca`, if `lambda` is:

Options

A. `-9,3`
B. 9,3
C. `3,-3`
D. None of these

Answer

To solve the problem step by step, we will analyze the given information and derive the value of \(\lambda\).

### Step 1: Understand the given information
We have two unit collinear vectors \(\vec{a}\) and \(\vec{c}\), which means:
- \(|\vec{a}| = 1\)
- \(|\vec{c}| = 1\)
- The angle between \(\vec{a}\) and \(\vec{c}\) is \(0^\circ\) (since they are collinear).

We are also given that \(|\vec{b}| = 6\).

### Step 2: Write the equation from the problem
The problem states:
\[ \vec{b} - 3\vec{c} = \lambda \vec{a} \]

### Step 3: Rearranging the equation
Rearranging the equation gives us:
\[ \vec{b} = \lambda \vec{a} + 3\vec{c} \]

### Step 4: Taking the magnitude of both sides
Taking the magnitude of both sides, we have:
\[ |\vec{b}| = |\lambda \vec{a} + 3\vec{c}| \]
Since \(|\vec{b}| = 6\), we can write:
\[ 6 = |\lambda \vec{a} + 3\vec{c}| \]

### Step 5: Using the properties of magnitudes
Using the formula for the magnitude of the sum of vectors, we can express this as:
\[ |\lambda \vec{a} + 3\vec{c}|^2 = |\lambda \vec{a}|^2 + |3\vec{c}|^2 + 2 |\lambda \vec{a}| |3\vec{c}| \cos(0^\circ) \]
Since \(|\vec{a}| = 1\) and \(|\vec{c}| = 1\), we have:
\[ |\lambda \vec{a}|^2 = \lambda^2 \]
\[ |3\vec{c}|^2 = 9 \]
Thus, we can rewrite the equation as:
\[ 36 = \lambda^2 + 9 + 6\lambda \]

### Step 6: Rearranging the equation
Rearranging gives us:
\[ \lambda^2 + 6\lambda + 9 - 36 = 0 \]
\[ \lambda^2 + 6\lambda - 27 = 0 \]

### Step 7: Solving the quadratic equation
Now we can solve the quadratic equation \(\lambda^2 + 6\lambda - 27 = 0\) using the factorization method:
\[ \lambda^2 + 9\lambda - 3\lambda - 27 = 0 \]
Factoring gives us:
\[ (\lambda + 9)(\lambda - 3) = 0 \]

### Step 8: Finding the values of \(\lambda\)
Setting each factor to zero gives us:
\[ \lambda + 9 = 0 \quad \Rightarrow \quad \lambda = -9 \]
\[ \lambda - 3 = 0 \quad \Rightarrow \quad \lambda = 3 \]

### Conclusion
Thus, the possible values of \(\lambda\) are \(-9\) and \(3\). The correct option from the choices provided is:
**A: \(-9, 3\)**.