Question

If A, B, C, D are any four points in space, then $| \overrightarrow { A B } \times \overrightarrow { C D } + \overrightarrow { B C } \times \overrightarrow { A D } + \overrightarrow { C A } \times \overrightarrow { B D } |$ is equal to

• A. $2 \Delta$
• B. $4 \Delta$
• C. $3 \Delta$
• D. (where ∆ denotes the area of $\triangle A B C$ )

Answer 1

Answer: (B)

$4 \Delta$

Answer 2

Let $A$ be the origin and let the poisition vectors of $B , C$ and $D$ be $\mathbf { b } , \mathbf { c }$ and d respectively.

Then $\overrightarrow { A B } = \mathbf { b }$ $\overrightarrow { C D } = \mathbf { d } - \mathbf { c }$ $\overrightarrow { B C } = \mathbf { c } - \mathbf { b }$ $\overrightarrow { A D } = \mathbf { d }$ $\overrightarrow { C A } = - \mathbf { c }$ and $\overrightarrow { B D } = \mathbf { d } - \mathbf { b }$

$\therefore | \overrightarrow { A B } \times \overrightarrow { C D } + \overrightarrow { B C } \times \overrightarrow { A D } + \overrightarrow { C A } \times \overrightarrow { B D } |$

$= | \mathbf { b } \times ( \mathbf { d } - \mathbf { c } ) + ( \mathbf { c } - \mathbf { b } ) \times \mathbf { d } - \mathbf { c } \times ( \mathbf { d } - \mathbf { b } ) |$

$= | \mathbf { b } \times \mathbf { d } - \mathbf { b } \times \mathbf { c } + \mathbf { c } \times \mathbf { d } - \mathbf { b } \times \mathbf { d } - \mathbf { c } \times \mathbf { d } + \mathbf { c } \times \mathbf { b } |$

$= 4$(area of triangle