Question

$A$: If $\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{c}\times \overrightarrow{d}$ and $\overrightarrow{a}\times \overrightarrow{c}=\overrightarrow{b}\times \overrightarrow{d}$ then $\overrightarrow{b}-\overrightarrow{c}$ is parallel to $\overrightarrow{a}-\overrightarrow{d}$

$R$: If cross product of two non-zero vectors is zero vector then those two vectors are parallel

• A. Both $A$ and $R$ are true and $R$ is the correct explanation of $A$
• B. Both $A$ and $R$ are true but $R$ is not correct explanation of $A$
• C. $A$ is true but $R$ is false
• D. $A$ is false but $R$ is true

Answer 1

The correct option is A Both $A$ and $R$ are true and $R$ is the correct explanation of $A$

Assertion $\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{c}\times \overrightarrow{d}----1$
$\overrightarrow{a}\times \overrightarrow{c}=\overrightarrow{b}\times \overrightarrow{d}----2$
subtract 1 and 2
$\overrightarrow{a}\times (\overrightarrow{b}-\overrightarrow{c})=(\overrightarrow{c}-\overrightarrow{b})\times \overrightarrow{d}$
$(\overrightarrow{a}-\overrightarrow{d})\times (\overrightarrow{b}-\overrightarrow{c})=0$
So $(\overrightarrow{a}-\overrightarrow{d})\parallel (\overrightarrow{b}-\overrightarrow{c})$