Question

If \(\vec{a}+2\vec{b}+3\vec{c}=\vec{0}\) and
\((\vec{a}\times\vec{b})+(\vec{b}\times\vec{c})+(\vec{c}\times\vec{a})=\lambda(\vec{b}\times\vec{c})\)
then the value of λ is equal to

• A. 4
• B. 2
• C. 6
• D. 3

Hint:

When solving vector equations involving cross products, remember to use the properties of cross products, such as \( \vec{u} \times \vec{u} = \vec{0} \) and \( \vec{a} \times \vec{b} = -\vec{b} \times \vec{a} \). These properties simplify the problem and make it easier to solve.

Answer 1

The Correct Option is C

The correct answer is: (C): 6

We are given the following vector equations:

\( \vec{a} + 2\vec{b} + 3\vec{c} = \vec{0} \)

\( (\vec{a} \times \vec{b}) + (\vec{b} \times \vec{c}) + (\vec{c} \times \vec{a}) = \lambda (\vec{b} \times \vec{c}) \)

Step 1: Express \( \vec{a} \) in terms of \( \vec{b} \) and \( \vec{c} \)

From the first equation, solve for \( \vec{a} \):

\( \vec{a} = -2\vec{b} - 3\vec{c} \)

Step 2: Substitute into the second equation

Now, substitute \( \vec{a} = -2\vec{b} - 3\vec{c} \) 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 3: Simplify the cross products

Now, simplify the terms involving the cross products. We know that \( \vec{u} \times \vec{u} = \vec{0} \), so \( \vec{b} \times \vec{b} = \vec{0} \) and \( \vec{c} \times \vec{c} = \vec{0} \). Therefore, we have:

\( (-2\vec{b} \times \vec{b}) - (3\vec{c} \times \vec{b}) + \vec{b} \times \vec{c} - 2(\vec{c} \times \vec{b}) - 3(\vec{c} \times \vec{c}) = \lambda (\vec{b} \times \vec{c}) \)

This simplifies to:

\( -3\vec{c} \times \vec{b} + \vec{b} \times \vec{c} - 2\vec{c} \times \vec{b} = \lambda (\vec{b} \times \vec{c}) \)

Step 4: Combine like terms

Now combine the terms involving \( \vec{c} \times \vec{b} \) and \( \vec{b} \times \vec{c} \):

\( -5\vec{c} \times \vec{b} + \vec{b} \times \vec{c} = \lambda (\vec{b} \times \vec{c}) \)

Since \( \vec{c} \times \vec{b} = -\vec{b} \times \vec{c} \), substitute this into the equation:

\( 5\vec{b} \times \vec{c} + \vec{b} \times \vec{c} = \lambda (\vec{b} \times \vec{c}) \)

Step 5: Factor the terms

Factor out \( \vec{b} \times \vec{c} \) on the left-hand side:

\( (5 + 1)(\vec{b} \times \vec{c}) = \lambda (\vec{b} \times \vec{c}) \)

Step 6: Solve for \( \lambda \)

Now, cancel out \( \vec{b} \times \vec{c} \) (assuming \( \vec{b} \times \vec{c} \neq \vec{0} \)) to get:

\( 6 = \lambda \)

Conclusion:
The value of \( \lambda \) is 6, so the correct answer is (C): 6.