Question

If \[\left[ 3 \vec{a} + 7 \vec{b} \vec{c} \vec{d} \right] = \lambda\left[ \vec{a} \vec{c} \vec{d} \right] + \mu\left[ \vec{b} \vec{c} \vec{d} \right],\] then find the value of λ + μ.

Answer 1

We have

\[\left[ 3 \vec{a} + 7 \vec{b} \vec{c} \vec{d} \right] = \lambda \left[ \vec{a} \vec{c} \vec{d} \right] + \mu\left[ \vec{b} \vec{c} \vec{d} \right]\]

\[ \Rightarrow \left[ \left( 3 \vec{a} + 7 \vec{b} \right) \times \vec{c} \right] . \vec{d} = \lambda \left[ \vec{a} \vec{c} \vec{d} \right] + \mu\left[ \vec{b} \vec{c} \vec{d} \right] \left(\text {  By definition of scalar triple product } \right)\]

\[ \Rightarrow \left[ \left( 3 \vec{a} \times \vec{c} \right) + \left( 7 \vec{b} \times \vec{c} \right) \right] . \vec{d} = \lambda \left[ \vec{a} \vec{c} \vec{d} \right] + \mu\left[ \vec{b} \vec{c} \vec{d} \right] \]

\[ \Rightarrow \left( 3 \vec{a} \times \vec{c} \right) . \vec{d} + \left( 7 \vec{b} \times \vec{c} \right) . \vec{d} = \lambda \left[ \vec{a} \vec{c} \vec{d} \right] + \mu\left[ \vec{b} \vec{c} \vec{d} \right] \]

\[ \Rightarrow \left[ 3 \vec{a} \vec{c} \vec{d} \right] + \left[ 7 \vec{b} \vec{c} \vec{d} \right] = \lambda \left[ \vec{a} \vec{c} \vec{d} \right] + \mu\left[ \vec{b} \vec{c} \vec{d} \right]\]

\[ \Rightarrow 3 \left[ \vec{a} \vec{c} \vec{d} \right] + 7 \left[ \vec{b} \vec{c} \vec{d} \right] = \lambda \left[ \vec{a} \vec{c} \vec{d} \right] + \mu\left[ \vec{b} \vec{c} \vec{d} \right] \left( \because \left[ \lambda \vec{a} \vec{b} \vec{c} \right] = \lambda\left[ \vec{a} \vec{b} \vec{c} \right] \text { for any scalar } \lambda \right)\]

Comparing both sides, we get

\[\lambda = 3 \]

\[\mu = 7\]

\[ \therefore \lambda + \mu = 3 + 7 = 10\]