In the following questions, a statement of assertion (Statement-1) is followed by a statement of reason (Statement-2). Mark the correct choice as :Statement-1 : Unit vectors orthogonal to the vector 3i^+2j^+6k^ and coplanar with the vectors 2i^+j^+k^ and i^-j^+k^ are ±110(3j^−k^).Statement-2 : For any three vectors a→,b→and c→, vector a→×(b→×c→) is orthogonal to a→and lies in the plane of b→and c→.

Statement-2 is true. The vector r→=a→×(b→×c→) is perpendicular to a→ and lies in the plane of b→ and c→.Let a→=3i^+2j^+6k^,b→=2i^+j^+k^ and c→=i^−j^+k^ Using statement-2, required unit vectors are given by α→=±a→×(b→×c→)|a→×(b→×c→)|Now, a→×(b→×c→)=(a→⋅c→)b→−(a→⋅b→)c→⇒ a→×(b→×c→)=7(2i^+j^+k^)−14(i^−j^+k^)=21j^−7k^∴ α→=±21j^−7k^710=±110(3j^−k^)So, statement-1 is true and statement-2 is a correct explanation of statement-1.

In the following questions, a statement of assertion (Statement-1) is followed by a statement of reason (Statement-2). Mark the correct choice as :Statement-1 : Unit vectors orthogonal to the vector 3i^+2j^+6k^ and coplanar with the vectors 2i^+j^+k^ and i^-j^+k^ are ±110(3j^−k^).Statement-2 : For any three vectors a→,b→and c→, vector a→×(b→×c→) is orthogonal to a→and lies in the plane of b→and c→.

Statement-2 is true. The vector r→=a→×(b→×c→) is perpendicular to a→ and lies in the plane of b→ and c→.Let a→=3i^+2j^+6k^,b→=2i^+j^+k^ and c→=i^−j^+k^ Using statement-2, required unit vectors are given by α→=±a→×(b→×c→)|a→×(b→×c→)|Now, a→×(b→×c→)=(a→⋅c→)b→−(a→⋅b→)c→⇒ a→×(b→×c→)=7(2i^+j^+k^)−14(i^−j^+k^)=21j^−7k^∴ α→=±21j^−7k^710=±110(3j^−k^)So, statement-1 is true and statement-2 is a correct explanation of statement-1.

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In the following questions, a statement of assertion (Statement-1) is followed by a statement of reason (Statement-2). Mark the correct choice as :
Statement-1 : Unit vectors orthogonal to the vector $3 \hat{i} + 2 \hat{j} + 6 \hat{k}$ and coplanar with the vectors $2 \hat{i} + \hat{j} + \hat{k}$ and $\hat{i} - \hat{j} + \hat{k}$ are $\pm \frac{1}{\sqrt{10}} (3 \hat{j} - \hat{k})$ .
Statement-2 : For any three vectors $\vec{a}, \vec{b}$ and $\vec{c}$ , vector $\vec{a} \times (\vec{b} \times \vec{c})$ is orthogonal to $\vec{a}$ and lies in the plane of $\vec{b}$ and $\vec{c}$ .

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If both Statement-1 and Statement-2 are true and Statement-2 is the correct explanation of Statement-1.

If both Statement-1 and Statement-2 are true but Statement-2 is not the correct explanation of Statement-1.

If Statement-1 is true but Statement-2 is false.

If Statement-1 is true but Statement-2 is true.

answer is A.

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Detailed Solution

Statement- $2$ is true.
The vector $\vec{r} = \vec{a} \times (\vec{b} \times \vec{c})$ is perpendicular to $\vec{a}$ and lies in the plane of $\vec{b}$ and $\vec{c}$ .
Let $\vec{a} = 3 \hat{i} + 2 \hat{j} + 6 \hat{k}, \vec{b} = 2 \hat{i} + \hat{j} + \hat{k}$ and $\vec{c} = \hat{i} - \hat{j} + \hat{k}$
Using statement- $2$ , required unit vectors are given by
$\vec{α} = \pm \frac{\vec{a} \times (\vec{b} \times \vec{c})}{| \vec{a} \times (\vec{b} \times \vec{c}) |}$
Now, $\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c}$
$\begin{array}{l} \Rightarrow \vec{a} \times (\vec{b} \times \vec{c}) = 7 (2 \hat{i} + \hat{j} + \hat{k}) - 14 (\hat{i} - \hat{j} + \hat{k}) = 21 \hat{j} - 7 \hat{k} \\ ∴ \vec{α} = \pm \frac{21 \hat{j} - 7 \hat{k}}{7 \sqrt{10}} = \pm \frac{1}{\sqrt{10}} (3 \hat{j} - \hat{k}) \end{array}$
So, statement- $1$ is true and statement- $2$ is a correct explanation of statement- $1$ .