if `vec(a) + 2vec(b) + 3vec(c ) = 0 and vec(a) xx vec(b) + vec(b) xx vec(c ) + vec(c ) xx vec(a) = lamda (vec(b) xx vec(c ))` then what is the value of `lamda`?

Options

A. A. 2
B. B. 3
C. C. 4
D. D. 6

Answer

To solve the problem, we will follow the steps outlined in the video transcript and derive the value of λ step by step.

### Step 1: Write the given equations
The problem states:
1. \(\vec{a} + 2\vec{b} + 3\vec{c} = 0\)
2. \(\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} = \lambda (\vec{b} \times \vec{c})\)

### Step 2: Express \(\vec{a}\) in terms of \(\vec{b}\) and \(\vec{c}\)
From the first equation, we can isolate \(\vec{a}\):
\[ \vec{a} = -2\vec{b} - 3\vec{c} \]

### Step 3: Substitute \(\vec{a}\) into the second equation
We substitute \(\vec{a}\) into the second equation:
\[ (-2\vec{b} - 3\vec{c}) \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times (-2\vec{b} - 3\vec{c}) = \lambda (\vec{b} \times \vec{c}) \]

### Step 4: Calculate each cross product
1. \((-2\vec{b}) \times \vec{b} = 0\) (since the cross product of any vector with itself is zero)
2. \((-3\vec{c}) \times \vec{b} = -3(\vec{c} \times \vec{b})\)
3. \(\vec{b} \times \vec{c}\) remains as is.
4. \(\vec{c} \times (-2\vec{b}) = -2(\vec{c} \times \vec{b})\)
5. \(\vec{c} \times (-3\vec{c}) = 0\) (again, the cross product of any vector with itself is zero)

Putting these together:
\[ 0 - 3(\vec{c} \times \vec{b}) + \vec{b} \times \vec{c} - 2(\vec{c} \times \vec{b}) + 0 = \lambda (\vec{b} \times \vec{c}) \]

### Step 5: Combine like terms
Now we can combine the terms:
\[ (-3 - 2)(\vec{c} \times \vec{b}) + \vec{b} \times \vec{c} = \lambda (\vec{b} \times \vec{c}) \]
This simplifies to:
\[ -5(\vec{c} \times \vec{b}) + \vec{b} \times \vec{c} = \lambda (\vec{b} \times \vec{c}) \]

### Step 6: Change the order of cross products
Using the property of cross products, we can rewrite \(\vec{c} \times \vec{b}\) as \(-(\vec{b} \times \vec{c})\):
\[ -5(-\vec{b} \times \vec{c}) + \vec{b} \times \vec{c} = \lambda (\vec{b} \times \vec{c}) \]
This simplifies to:
\[ 5(\vec{b} \times \vec{c}) + \vec{b} \times \vec{c} = \lambda (\vec{b} \times \vec{c}) \]

### Step 7: Factor out \(\vec{b} \times \vec{c}\)
Combining the terms gives:
\[ (5 + 1)(\vec{b} \times \vec{c}) = \lambda (\vec{b} \times \vec{c}) \]
Thus, we have:
\[ 6(\vec{b} \times \vec{c}) = \lambda (\vec{b} \times \vec{c}) \]

### Step 8: Solve for \(\lambda\)
Assuming \(\vec{b} \times \vec{c} \neq 0\), we can divide both sides by \(\vec{b} \times \vec{c}\):
\[ \lambda = 6 \]

### Final Answer
The value of \(\lambda\) is:
\[ \lambda = 6 \]