Question

If $\hat{\alpha }=p(\hat{b}\times \hat{c})+q(\hat{c}\times \hat{a})+r(\hat{a}\times \hat{b})$ and if $\hat{\alpha }.(\hat{a}+\hat{b}+\hat{c})=1$, then $\left[\hat{a}\hat{b}\hat{c}\right]=$

• A. $p+q+r$
• B. $\frac{1}{p+q+r}$
• C. $2(p+q+r)$
• D. $\frac{2}{p+q+r}$

Answer 1

The correct option is B $\frac{1}{p+q+r}$

$\hat{\alpha }=p(\hat{b}\times \hat{c})+q(\hat{c}\times \hat{a})+r(\hat{a}\times \hat{b})$

$\hat{\alpha }.(\hat{a}+\hat{b}+\hat{c})=1$

$\left[\hat{a}\hat{b}\hat{c}\right]=?$

$\hat{a}.(\hat{b}\times \hat{c})=?$

$p(\hat{b}\times \hat{c})+q(\hat{c}\times \hat{a})+r(\hat{a}\times \hat{b}).(\hat{a}+\hat{b}+\hat{c})=1$

$p[\hat{a}.(\hat{b}\times \hat{c}\left)\right]+q[\hat{b}.(\hat{c}\times \hat{a}\left)\right]+r[\hat{c}.(\hat{a}\times \hat{b}\left)\right]=1$

$[p+q+r]\left[\hat{a}\hat{b}\hat{c}\right]=1$

$\left[\hat{a}\hat{b}\hat{c}\right]=\frac{1}{p+q+r}$

Hence option B is correct.