Browse Questions

If $\overrightarrow a \times \overrightarrow b=\overrightarrow c \times \overrightarrow d\: and \: \overrightarrow a \times \overrightarrow c=\overrightarrow b \times \overrightarrow d$ show that $\overrightarrow a-\overrightarrow d$ is parallel to $\overrightarrow b-\overrightarrow c$ where $\overrightarrow a \neq \overrightarrow d\: and \overrightarrow b \neq\overrightarrow c$

Toolbox:
• $\overrightarrow a\times\overrightarrow a = \overrightarrow 0$
• $\overrightarrow a\times \overrightarrow b= -(\overrightarrow b\times \overrightarrow a)$
• If two vectors are parallel then their cross product is zero
Step 1:
Given :
$\overrightarrow a\times \overrightarrow b=\overrightarrow c\times \overrightarrow d$
$\overrightarrow a\times \overrightarrow c=\overrightarrow b\times \overrightarrow d$
If two vectors are parallel then their cross product is zero
If $(\overrightarrow a-\overrightarrow d)$ is parallel to $(\overrightarrow b-\overrightarrow c)$ then $(\overrightarrow a-\overrightarrow d)\times (\overrightarrow b-\overrightarrow c)$
$\Rightarrow (\overrightarrow a\times \overrightarrow b)-(\overrightarrow a\times \overrightarrow c)-(\overrightarrow d\times \overrightarrow b)+(\overrightarrow d\times \overrightarrow c)$
Step 2:
But $\overrightarrow a\times \overrightarrow c=\overrightarrow b\times \overrightarrow d$
and $\overrightarrow d\times \overrightarrow b=-(\overrightarrow b\times \overrightarrow d)$
$\overrightarrow a\times \overrightarrow b=\overrightarrow c\times \overrightarrow d$
$(\overrightarrow d\times \overrightarrow c)=-(\overrightarrow c\times \overrightarrow d)$
Substituting this in the above step,we get
$(\overrightarrow c\times \overrightarrow d)-(\overrightarrow b\times \overrightarrow d)+(\overrightarrow b\times \overrightarrow d)-(\overrightarrow c\times \overrightarrow d)$
$\Rightarrow 0$
Hence $(\overrightarrow a-\overrightarrow d)$ is parallel to $(\overrightarrow b-\overrightarrow c)$