If \(\hat{p}, \hat{q}, \hat{r}\) are mutually perpendicular unit vectors and form a right handed triplet then the value of \(\hat{p}\cdot(\hat{q}\times\hat{r}) + \hat{q}\cdot(\hat{p}\times\hat{r}) + \hat{r}\cdot(\hat{p}\times\hat{q}) =\) _____

Q.1 (iii) If \(\hat{p}, \hat{q}, \hat{r}\) are mutually perpendicular unit vectors and form a right handed triplet then the value of \(\hat{p}\cdot(\hat{q}\times\hat{r}) + \hat{q}\cdot(\hat{p}\times\hat{r}) + \hat{r}\cdot(\hat{p}\times\hat{q}) =\) _____

A. \(-2\)

B. \(0\)

C. \(1\)

D. \(3\)

Doubtly Asked 28 Oct 2026 Board exam · June 2026 hsc 12th Save WhatsApp Report

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Answer: C. \(1\)

Answer: (c) 1

For a right handed triplet of mutually perpendicular unit vectors, \([\hat{p}\ \hat{q}\ \hat{r}] = 1\).

\(\hat{p}\cdot(\hat{q}\times\hat{r}) = [\hat{p}\ \hat{q}\ \hat{r}] = 1\)
\(\hat{q}\cdot(\hat{p}\times\hat{r}) = [\hat{q}\ \hat{p}\ \hat{r}] = -[\hat{p}\ \hat{q}\ \hat{r}] = -1\) (one interchange)
\(\hat{r}\cdot(\hat{p}\times\hat{q}) = [\hat{r}\ \hat{p}\ \hat{q}] = [\hat{p}\ \hat{q}\ \hat{r}] = 1\) (cyclic order)

Sum \(= 1 - 1 + 1 = 1\).

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If \(\hat{p}, \hat{q}, \hat{r}\) are mutually perpendicular unit vectors and form a right handed triplet then the value of \(\hat{p}\cdot(\hat{q}\times\hat{r}) + \hat{q}\cdot(\hat{p}\times\hat{r}) + \hat{r}\cdot(\hat{p}\times\hat{q}) =\) _____

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