The GATE Grind

GATE 2024 CE (CE2) – Question 44

Engineering Mathematics · Calculus: Integrals, partial and total derivatives, gradient, divergence, curl, line, surface and volume integrals · 2 marks · Multiple select

Three vectors $\vec{p}$, $\vec{q}$, and $\vec{r}$ are given as
$\vec{p} = \hat{i} + \hat{j} + \hat{k}$, $\vec{q} = \hat{i} + 2\hat{j} + 3\hat{k}$, $\vec{r} = 2\hat{i} + 3\hat{j} + 4\hat{k}$
Which of the following is/are CORRECT?

  1. $\vec{p} \times (\vec{q} \times \vec{r}) + \vec{q} \times (\vec{r} \times \vec{p}) + \vec{r} \times (\vec{p} \times \vec{q}) = \vec{0}$
  2. $\vec{p} \times (\vec{q} \times \vec{r}) = (\vec{p} \cdot \vec{r})\vec{q} - (\vec{p} \cdot \vec{q})\vec{r}$
  3. $\vec{p} \times (\vec{q} \times \vec{r}) = (\vec{p} \times \vec{q}) \times \vec{r}$
  4. $\vec{r} \cdot (\vec{p} \times \vec{q}) = (\vec{q} \times \vec{p}) \cdot \vec{r}$

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Show answer and explanation

Correct answer: (A) $\vec{p} \times (\vec{q} \times \vec{r}) + \vec{q} \times (\vec{r} \times \vec{p}) + \vec{r} \times (\vec{p} \times \vec{q}) = \vec{0}$; (B) $\vec{p} \times (\vec{q} \times \vec{r}) = (\vec{p} \cdot \vec{r})\vec{q} - (\vec{p} \cdot \vec{q})\vec{r}$; (D) $\vec{r} \cdot (\vec{p} \times \vec{q}) = (\vec{q} \times \vec{p}) \cdot \vec{r}$

Explanation

A is the Jacobi identity, which holds for any three vectors. B is the vector triple product (BAC-CAB) rule, which is true. C is false in general: here $\vec{q} \times \vec{r} = (-1, 2, -1)$ and $\vec{p} \times (\vec{q} \times \vec{r}) = (-3, 0, 3)$, while $(\vec{p} \times \vec{q}) \times \vec{r} = (1, -2, 1) \times (2, 3, 4) = (-11, -2, 7)$. D: $(\vec{q} \times \vec{p}) \cdot \vec{r} = -\vec{r} \cdot (\vec{p} \times \vec{q})$, so D holds only if the scalar triple product is zero. Here $\vec{p} \cdot (\vec{q} \times \vec{r}) = (1)(-1) + (1)(2) + (1)(-1) = 0$ (the three vectors are coplanar), so D is true for these vectors.