If `[(2veca+vecb)veccvecd]=lambda[vecaveccvecd]+mu[vecbveccvecd]` then `lambda+mu` is equal to

Options

A. `-6`
B. 3
C. 2
D. `-2`

Answer

To solve the problem, we need to analyze the equation given in the question:

\[ [(2\vec{a} + \vec{b}) \cdot (\vec{c} \times \vec{d})] = \lambda [\vec{a} \cdot (\vec{c} \times \vec{d})] + \mu [\vec{b} \cdot (\vec{c} \times \vec{d})] \]

### Step 1: Expand the Left Side
We start by expanding the left side using the properties of the scalar triple product:

\[ (2\vec{a} + \vec{b}) \cdot (\vec{c} \times \vec{d}) = 2(\vec{a} \cdot (\vec{c} \times \vec{d})) + (\vec{b} \cdot (\vec{c} \times \vec{d})) \]

### Step 2: Rewrite the Left Side
Now we can rewrite the left side in terms of the scalar triple products:

\[ = 2[\vec{a} \cdot (\vec{c} \times \vec{d})] + [\vec{b} \cdot (\vec{c} \times \vec{d})] \]

### Step 3: Set Up the Equation
Now we can set this equal to the right side of the original equation:

\[ 2[\vec{a} \cdot (\vec{c} \times \vec{d})] + [\vec{b} \cdot (\vec{c} \times \vec{d})] = \lambda [\vec{a} \cdot (\vec{c} \times \vec{d})] + \mu [\vec{b} \cdot (\vec{c} \times \vec{d})] \]

### Step 4: Compare Coefficients
Now we can compare the coefficients of the scalar triple products on both sides:

1. For \(\vec{a} \cdot (\vec{c} \times \vec{d})\):
\[ 2 = \lambda \]

2. For \(\vec{b} \cdot (\vec{c} \times \vec{d})\):
\[ 1 = \mu \]

### Step 5: Find \(\lambda + \mu\)
Now we can find \(\lambda + \mu\):

\[ \lambda + \mu = 2 + 1 = 3 \]

### Final Answer
Thus, the value of \(\lambda + \mu\) is:

\[ \boxed{3} \]