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If [ 2 vec (a) - 3 vec(b) vec( c ) v...

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=`

A

13

B

`-5`

C

7

D

none of these

Text Solution

AI Generated Solution

The correct Answer is:
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 \(-5\).
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