Question

Statement-I : $\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c}),\overrightarrow{b}\times (\overrightarrow{c}\times \overrightarrow{a}),\overrightarrow{c}\times (\overrightarrow{a}\times \overrightarrow{b})$ are coplanar vectors.

Statement-II: If there exists scalars $l,m,n$ not all zero such that $l\overrightarrow{a}+m\overrightarrow{b}+n\overrightarrow{c}=\overrightarrow{0}$, then the vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are coplanar

• A. Both I and II are true and II is the correct explanation of I
• B. Both I and II are true but II is not correct explanation of I.
• C. I is true, II is false
• D. I is false, II is true

Answer 1

Answer: (A)

Both I and II are true and II is the correct explanation of I

When we look at statement 2, $l\overline{a}+m\overline{b}+n\overline{c}=0$ implies any of the three vectors $\overline{a},\overline{b}$ or $\overline{c}$ can be written in terms of the remaining two vectors.

Thus, the vectors become coplanar and hence statement 2 is true.

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

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

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

Since dot product is commutative, when we add the three relations, we get

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

$=(\overline{a}\cdot \overline{c})\overline{b}-(\overline{a}\cdot \overline{b})\overline{c}+(\overline{a}.\overline{b})\overline{c}-(\overline{b}.\overline{c})\overline{a}+(\overline{b}\cdot \overline{c})\overline{a}-(\overline{a}\cdot \overline{c})\overline{b}$

$=0$

Therefore the vectors $\overline{a}\times (\overline{b}\times \overline{c}),\overline{b}\times (\overline{c}\times \overline{a}),\overline{c}\times (\overline{a}\times \overline{b})$ are coplanar and thus it becomes similar to the explanation provided in statement 2.

Hence, both I and II are correct and II is the correct explanation of I.