If $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar vectors and $\lambda$ is a real number then $\left[\lambda(\vec{a}+\vec{b}) \lambda^2 \vec{b} \lambda \vec{c}\right]=[\vec{a} \vec{b}+\vec{c} \vec{b}]$ for

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Aicharya · Accepted Answer

$ \begin{aligned} & {\left[\begin{array}{lll} \lambda(\vec{a}+\vec{b}) & \lambda^2 \vec{b} & \lambda \vec{c} \end{array}ight]=\left[\begin{array}{lll} \vec{a} & \vec{b}+\vec{c} & \vec{b} \end{array}ight]} \ & \left|\begin{array}{ccc} \lambda & \lambda & 0 \ 0 & \lambda^2 & 0 \ 0 & 0 & \lambda \end{array}ight|=\left|\begin{array}{lll} 1 & 0 & 0 \ 0 & 1 & 1 \ 0 & 1 & 0 \end{array}ight| \ & \Rightarrow \lambda^4=-1 \ & \end{aligned} $ Hence no real value of $\lambda$

If $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar vectors and $\lambda$ is a real number then…

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