If `[ 2 vec (a) - 3 vec(b) vec( c ) vec(d)] =lambda [vec(a) vec(c ) vec(d) ] + mu [ vec(b) vec( c ) vec( d) ] `, then `2 lambda + 3 mu=`

Options

A. 13
B. `-5`
C. 7
D. none of these

Answer

To solve the problem, we need to analyze the given equation involving vectors and scalar triple products. The equation is:

\[ [2 \vec{a} - 3 \vec{b} \quad \vec{c} \quad \vec{d}] = \lambda [\vec{a} \quad \vec{c} \quad \vec{d}] + \mu [\vec{b} \quad \vec{c} \quad \vec{d}] \]

### Step-by-step Solution:

1. **Understanding Scalar Triple Product**:
The scalar triple product of vectors \(\vec{a}, \vec{b}, \vec{c}\) can be expressed as:
\[ [\vec{a} \quad \vec{b} \quad \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c}) \]
This means that the left-hand side and the right-hand side of the equation represent scalar triple products.

2. **Expanding the Left Side**:
We can rewrite the left-hand side:
\[ [2 \vec{a} - 3 \vec{b} \quad \vec{c} \quad \vec{d}] = [2 \vec{a} \quad \vec{c} \quad \vec{d}] - [3 \vec{b} \quad \vec{c} \quad \vec{d}] \]
This can be expressed as:
\[ 2(\vec{a} \cdot (\vec{c} \times \vec{d})) - 3(\vec{b} \cdot (\vec{c} \times \vec{d})) \]

3. **Expanding the Right Side**:
The right-hand side can be expanded as:
\[ \lambda [\vec{a} \quad \vec{c} \quad \vec{d}] + \mu [\vec{b} \quad \vec{c} \quad \vec{d}] = \lambda (\vec{a} \cdot (\vec{c} \times \vec{d})) + \mu (\vec{b} \cdot (\vec{c} \times \vec{d})) \]

4. **Setting the Two Sides Equal**:
Now, we can set the two sides equal to each other:
\[ 2(\vec{a} \cdot (\vec{c} \times \vec{d})) - 3(\vec{b} \cdot (\vec{c} \times \vec{d})) = \lambda (\vec{a} \cdot (\vec{c} \times \vec{d})) + \mu (\vec{b} \cdot (\vec{c} \times \vec{d})) \]

5. **Comparing Coefficients**:
From the equation, we can compare coefficients of \(\vec{a} \cdot (\vec{c} \times \vec{d})\) and \(\vec{b} \cdot (\vec{c} \times \vec{d})\):
- For \(\vec{a}\):
\[ 2 = \lambda \]
- For \(\vec{b}\):
\[ -3 = \mu \]

6. **Finding \(2\lambda + 3\mu\)**:
Now substituting the values of \(\lambda\) and \(\mu\):
\[ 2\lambda + 3\mu = 2(2) + 3(-3) = 4 - 9 = -5 \]

### Final Answer:
Thus, the value of \(2\lambda + 3\mu\) is \(-5\).