If $\overline{\mathrm{a}}, \overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ are unit coplanar vectors, then the scalar triple product $\left[\begin{array}{lll}2 \bar{a}-\bar{b} & 2 \bar{b}-\bar{c} & 2 \bar{c}-\bar{a}\end{array}\right]$ has the value

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Aicharya · Accepted Answer

$\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}$ are coplanar vectors. $\Rightarrow\left[\begin{array}{lll} \overline{\mathrm{a}} & \overline{\mathrm{~b}} & \overline{\mathrm{c}} \end{array}ight]=0$ Let $\bar{\alpha}=2 \overline{\mathrm{a}}-\overline{\mathrm{b}}, \bar{\beta}=2 \overline{\mathrm{~b}}-\overline{\mathrm{c}}$ and $\gamma=2 \overline{\mathrm{c}}-\overline{\mathrm{a}}$. Then, $\begin{aligned} & {\left[\begin{array}{ccc} \bar{\alpha} & \bar{\beta} & \bar{\gamma} \end{array}ight]=\left|\begin{array}{ccc} 2 & -1 & 0 \ 0 & 2 & -1 \ -1 & 0 & 2 \end{array}ight|\left[\begin{array}{lll} \overline{\mathrm{a}} & \overline{\mathrm{~b}} & \overline{\mathrm{c}} \end{array}ight]} \ & \Rightarrow\left[\begin{array}{lll} \bar{\alpha} & \bar{\beta} & \bar{\gamma} \end{array}ight]=7\left[\begin{array}{lll} \overline{\mathrm{a}} & \overline{\mathrm{~b}} & \overline{\mathrm{c}} \end{array}ight]=7(0)=0 \end{aligned}$

If $\overline{\mathrm{a}}, \overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ are unit coplanar vectors,…

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