If $\vec{u}, \vec{v}$ and $\vec{w}$ are three non-coplanar vectors, then $(\vec{u}+\vec{v}-\vec{w}) …

If $\vec{u}, \vec{v}$ and $\vec{w}$ are three non-coplanar vectors, then $(\vec{u}+\vec{v}-\vec{w}) .(\vec{u}-\vec{v}) \times(\vec{v}-\vec{w})$ equals
  1. $3 \vec{u} . \vec{v} \times \vec{w}$
  2. 0
  3. $\overrightarrow{\mathrm{u}} \cdot \overrightarrow{\mathrm{v}} \times \overrightarrow{\mathrm{w}}$
  4. $\overrightarrow{\mathrm{u}} \cdot \overrightarrow{\mathrm{w}} \times \overrightarrow{\mathrm{v}}$

Solution

$(\overrightarrow{\mathrm{u}}+\overrightarrow{\mathrm{v}}-\overrightarrow{\mathrm{w}}) \cdot(\overrightarrow{\mathrm{u}} \times \overrightarrow{\mathrm{v}}-\overrightarrow{\mathrm{u}} \times \overrightarrow{\mathrm{w}}-\overrightarrow{\mathrm{v}} \times \overrightarrow{\mathrm{v}}+\overrightarrow{\mathrm{v}} \times \overrightarrow{\mathrm{w}})$ $(\overrightarrow{\mathrm{u}}+\overrightarrow{\mathrm{v}}-\overrightarrow{\mathrm{w}}) \cdot(\overrightarrow{\mathrm{u}} \times \overrightarrow{\mathrm{v}}-\overrightarrow{\mathrm{u}} \times \overrightarrow{\mathrm{w}}+\overrightarrow{\mathrm{v}} \times \overrightarrow{\mathrm{w}})=\frac{\overrightarrow{\mathrm{u}} \cdot(\overrightarrow{\mathrm{u}} \times \overrightarrow{\mathrm{v}})}{0}$ $-\frac{\overrightarrow{\mathrm{u}} .(\overrightarrow{\mathrm{u}} \times \overrightarrow{\mathrm{w}})}{0}+\overrightarrow{\mathrm{u}} .(\overrightarrow{\mathrm{v}} \times \overrightarrow{\mathrm{w}})+\frac{\overrightarrow{\mathrm{v}} \cdot(\overrightarrow{\mathrm{u}} \times \overrightarrow{\mathrm{v}})}{0}-\overrightarrow{\mathrm{v}} \cdot(\overrightarrow{\mathrm{u}} \times \overrightarrow{\mathrm{w}})$ $+\frac{\overrightarrow{\mathrm{v}} \cdot(\overrightarrow{\mathrm{v}} \times \overrightarrow{\mathrm{w}})}{0}-\overrightarrow{\mathrm{w}} \cdot(\overrightarrow{\mathrm{u}} \times \overrightarrow{\mathrm{v}})+\frac{\overrightarrow{\mathrm{w}} \cdot(\overrightarrow{\mathrm{u}} \times \overrightarrow{\mathrm{w}})}{0}-\frac{\overrightarrow{\mathrm{w}} \cdot(\overrightarrow{\mathrm{u}} \times \overrightarrow{\mathrm{w}})}{0}=\overrightarrow{\mathrm{u}} \cdot(\overrightarrow{\mathrm{v}} \times \overrightarrow{\mathrm{w}})-\overrightarrow{\mathrm{v}} \cdot(\overrightarrow{\mathrm{u}} \times \overrightarrow{\mathrm{w}})-\overrightarrow{\mathrm{w}} \cdot(\overrightarrow{\mathrm{u}} \times \overrightarrow{\mathrm{v}})$ $=[\overrightarrow{\mathrm{u}} \overrightarrow{\mathrm{v}} \overrightarrow{\mathrm{w}}]+[\overrightarrow{\mathrm{v}} \overrightarrow{\mathrm{w}} \overrightarrow{\mathrm{u}}]-[\overrightarrow{\mathrm{w}} \overrightarrow{\mathrm{u}} \overrightarrow{\mathrm{v}}]=\overrightarrow{\mathrm{u}} .(\overrightarrow{\mathrm{v}} \times \overrightarrow{\mathrm{w}})$

Asked in: JEE Main 2003

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