Question

The vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ and $\overrightarrow{d}$ are such that $\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{c}\times \overrightarrow{d}$ and $\overrightarrow{a}\times \overrightarrow{c}=\overrightarrow{b}\times \overrightarrow{d}$. Which of the following is\are correct?
1. $(\overrightarrow{a}-\overrightarrow{d})\times (\overrightarrow{b}-\overrightarrow{c})=\overrightarrow{0}$
2. $(\overrightarrow{a}\times \overrightarrow{d})\times (\overrightarrow{b}\times \overrightarrow{c})=\overrightarrow{0}$
Select the correct answer using the code given below :

• A. 1 only
• B. 2 only
• C. Both 1 and 2
• D. Neither 1 nor 2

Answer 1

The correct option is A 1 only

Solution:

Given that:

$\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{c}\times \overrightarrow{d}$ and $\overrightarrow{a}\times \overrightarrow{c}=\overrightarrow{b}\times \overrightarrow{d}$

$1.(\overrightarrow{a}-\overrightarrow{d})\times (\overrightarrow{b}-\overrightarrow{c})$

$=\overrightarrow{a}\times \overrightarrow{b}-\overrightarrow{a}\times \overrightarrow{c}$$-\overrightarrow{d}\times \overrightarrow{b}+\overrightarrow{d}\times \overrightarrow{c}$

$=\overrightarrow{c}\times \overrightarrow{d}-\overrightarrow{b}\times \overrightarrow{d}$$+\overrightarrow{b}\times \overrightarrow{d}-\overrightarrow{c}\times \overrightarrow{d}$

$=0$

Statement 1 is correct.

$2.(\overrightarrow{a}\times \overrightarrow{d})\times (\overrightarrow{b}\times \overrightarrow{c})$

$=\left[\right\{(\overrightarrow{a}\times \overrightarrow{d}).\overrightarrow{c}\left\}\overrightarrow{b}\right]$$-\left[\right\{(\overrightarrow{a}\times \overrightarrow{d}).\overrightarrow{b}\left\}\overrightarrow{c}\right]$

$\ne 0$

Statement 2 is incorrect.

Hence, A is the correct option.