Question

Prove the following equation , image pasted below

Answer 1

Step-by-step explanation:

We have,

$\left[ \begin{array}{r}\vec{a} \times \vec{b} & \vec{b} \times \vec{c} & \vec{c} \times \vec{a} \end{array}\right]$

$= \left(\vec{a} \times \vec{b} \right) \cdot \left \{ \left(\vec{b} \times \vec{c} \right) \times \left(\vec{c} \times \vec{a} \right) \right \}$

$= \left(\vec{a} \times \vec{b} \right) \cdot \left[ \left \{ \left(\vec{b} \times \vec{c} \right) \cdot \vec{a} \right \} \vec{c} - \left \{ \left(\vec{b} \times \vec{c} \right) \cdot \vec{c} \right \} \vec{a}\right]$

$= \left(\vec{a} \times \vec{b} \right) \cdot \left \{ \left[ \vec{a} \: \: \vec{b} \: \: \vec{c} \right] \vec{c} - 0 \right \}$

$= \left(\vec{a} \times \vec{b} \right) \cdot \left \{ \left[\begin{array}{r} \vec{a} & \vec{b} & \vec{c} \end{array}\right] \vec{c} - 0 \right \}$

$= \left[\begin{array}{r} \vec{a} & \vec{b} & \vec{c} \end{array}\right] \left(\vec{a} \times \vec{b} \right) \cdot \vec{c}$

$= \left[\begin{array}{r} \vec{a} & \vec{b} & \vec{c} \end{array}\right] \cdot \left[\begin{array}{r} \vec{a} & \vec{b} & \vec{c} \end{array}\right]$

$= \left[\begin{array}{r} \vec{a} & \vec{b} & \vec{c} \end{array}\right]^{2}$