Consider the cube in the first octant with sides OP, OQ and OR of length 1, along the x-axis, y-axis and z-axis, respectively, where O(0,0,0) is the origin. Let S12,12,12 be the centre of the cube and T be the vertex of the cube opposite to the origin O such that S lies on the diagonal OT. If p→=SP→, q→=SQ→, r→=SR→ and t→=ST→, then the value of p→×q→×r→×t→ is ______.

JEE Advanced · Mathematics · 30. Vector Algebra

Consider the cube in the first octant with sides $O P, O Q$ and OR of length $1$ , along the x-axis, y-axis and z-axis, respectively, where $O (0, 0, 0)$ is the origin. Let $S (\frac{1}{2}, \frac{1}{2}, \frac{1}{2})$ be the centre of the cube and $T$ be the vertex of the cube opposite to the origin $O$ such that $S$ lies on the diagonal $O T$ . If $\vec{p} = \vec{S P}, \vec{q} = \vec{S Q}, \vec{r} = \vec{S R}$ and $\vec{t} = \vec{S T}$ , then the value of $|(\vec{p} \times \vec{q}) \times (\vec{r} \times \vec{t})|$ is ______.

A 0.51
B 0.7
C 0.5
D 1

Verified Solution

Answer & Solution

Correct Answer

(C) 0.5

Step-by-step Solution

Detailed explanation

\vec{p} = \vec{S P} = (\frac{1}{2}, - \frac{1}{2}, - \frac{1}{2}) = \frac{1}{2} (\hat{i} - \hat{j} - \hat{k})

\vec{q} = \vec{S Q} = (- \frac{1}{2}, \frac{1}{2}, - \frac{1}{2}) = \frac{1}{2} (- \hat{i} + \hat{j} - \hat{k})

\vec{r} = \vec{S R} = (- \frac{1}{2}, - \frac{1}{2}, \frac{1}{2}) = \frac{1}{2} (- \hat{i} - \hat{j} + \hat{k})

\vec{t} = \vec{S T} = (\frac{1}{2}, \frac{1}{2}, \frac{1}{2}) = \frac{1}{2} (\hat{i} + \hat{j} + \hat{k})

|(\vec{p} \times \vec{q}) \times (\vec{r} \times \vec{t})| = |\frac{1}{4} |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & - 1 & - 1 \\ - 1 & 1 & - 1 \end{matrix}| \times \frac{1}{4} |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ - 1 & - 1 & 1 \\ 1 & 1 & 1 \end{matrix}||

= \frac{1}{16} |(2 \hat{i} + 2 \hat{j}) \times (- 2 \hat{i} + 2 \hat{j})| = |\frac{\hat{k}}{2}| = \frac{1}{2}

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Atomic number	Ionization Enthalpy (kJ/mol)
	$I_{1}$	$I_{2}$	$I_{3}$
$n$	$1681$	$3374$	$6050$
$n + 1$	$2081$	$3952$	$6122$
$n + 2$	$496$	$4562$	$6910$
$n + 3$	$738$	$1451$	$7733$