Question

If the vectors $PQ,QR,RS,ST,TU,$ and $UP$ represent the sides of a regular hexagon. Then, Statement I. $PQ\times \left(RS+ST\right)\ne 0$ because Statement II. $PQ\times RS=0$ and $PQ\times ST\ne 0$.

• A. Statement I is correct, Statement II is correct; Statement II is correct explanation for Statement I
• B. Statement I is correct, Statement II is correct; Statement II is not correct explanation for Statement I
• C. Statement I is correct; Statement II is incorrect
• D. Statement I is incorrect; Statement II is correct

Answer 1

The correct option is C

Statement I is correct; Statement II is incorrect

Explanation for the correct option :

Checking both the statements,

The six sides of a regular hexagon is represented by vectors as $PQ,QR,RS,ST,TU,UP$ and it can be shown as:

Now the statement I is$PQ\times \left(RS+ST\right)\ne 0$.

The left hand side of the statement can be split as:

$PQ\times \left(RS+ST\right)=PQ\times RS+PQ\times ST$

Now, vectors $PQ$ and $RS$ are not parallel so $PQ\times RS\ne 0$ , so the first term is not equal to zero.

The vectors $PQ$ and $ST$ are parallel so $PQ\times ST=0$, so the second term is equal to zero.

So overall the expression $PQ\times RS+PQ\times ST$ is not equal to zero. So the statement $PQ\times \left(RS+ST\right)\ne 0$ is true.

It is already shown that $PQ\times RS\ne 0$ and $PQ\times ST=0$, so the statement II stating that $PQ\times RS=0$ and $PQ\times ST\ne 0$ is false.

Thus, the first statement is correct and the second statement is incorrect.

Hence, the correct option is C.