Question

If $\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{c}\times \overrightarrow{d}$ and $\overrightarrow{a}\times \overrightarrow{c}=\overrightarrow{b}\times \overrightarrow{d}$, then the vectors $\overrightarrow{a}-\overrightarrow{d}$ and $\overrightarrow{b}-\overrightarrow{c}$ are

• A. collinear
• B. linearly independent
• C. perpendicular
• D. parallel

Answer 1

The correct option is D parallel

$\overline{a}-\overline{d},\overline{b}-\overline{c}$

$Crossproductof\overline{a}-\overline{d},\overline{b}-\overline{c}is:$

$⟹(\overline{a}-\overline{d})\times (\overline{b}-\overline{c})$

$=(\overline{a}\times \overline{b})-(\overline{a}\times \overline{c})-(\overline{d}\times \overline{b})+(\overline{d}\times \overline{c})$

$=(\overline{a}\times \overline{b})-(\overline{a}\times \overline{c})+(\overline{b}\times \overline{d})-(\overline{c}\times \overline{d})-\left(i\right)$

$(\because \overline{a}\times \overline{b}=-(\overline{b}\times \overline{a}\left)\right)$

$(\because \overline{a}\times \overline{b}=\overline{c}\times \overline{d}and\overline{a}\times \overline{c}=\overline{b}\times \overline{d})Puttingin\left(i\right)$

$⟹(\overline{a}\times \overline{b})-(\overline{a}\times \overline{c})+(\overline{a}\times \overline{c})-(\overline{a}\times \overline{b})$

$=0$

$\because Crossproductof\overline{a}-\overline{d}\&\overline{b}-\overline{c}=0,$

$(\because \sin \theta =0⟹\theta =0)$

$when\theta isanglebetweenthetwovectors.$