Let \overset{\rightarrow}{a} , \overset{\rightarrow}{b} \textrm{ }\textrm{ } and \textrm{ }\textrm{ } \overset{\rightarrow}{c} be three unit vectors and \overset{\rightarrow}{p} , \overset{\rightarrow}{q} , \overset{\rightarrow}{r} be three vectors such that \overset{\rightarrow}{p} = \left(\right. \overset{\rightarrow}{a} \cdot \overset{\rightarrow}{b} \left.\right) \textrm{ } \overset{\rightarrow}{a} - \overset{\rightarrow}{b} , \textrm{ }\textrm{ } \overset{\rightarrow}{q} = \left(\right. \overset{\rightarrow}{b} \cdot \overset{\rightarrow}{c} \left.\right) \textrm{ } \overset{\rightarrow}{b} - \overset{\rightarrow}{c} \textrm{ }\textrm{ } and \textrm{ }\textrm{ } \overset{\rightarrow}{r} = \left(\right. \overset{\rightarrow}{c} \cdot \overset{\rightarrow}{a} \left.\right) \textrm{ } \overset{\rightarrow}{c} - \overset{\rightarrow}{a} . If \overset{\rightarrow}{a} + \overset{\rightarrow}{b} + \overset{\rightarrow}{c} = 0 , then which of the following is/are correct ?

Engineering

Mathematics

Vector Triple Product

Question

Let $\vec{a}, \vec{b} and \vec{c}$ be three unit vectors and $\vec{p}, \vec{q}, \vec{r}$ be three vectors such that $\vec{p} = (\vec{a} \cdot \vec{b}) \vec{a} - \vec{b}, \vec{q} = (\vec{b} \cdot \vec{c}) \vec{b} - \vec{c} and \vec{r} = (\vec{c} \cdot \vec{a}) \vec{c} - \vec{a}$ . If $\vec{a} + \vec{b} + \vec{c} = 0,$

then which of the following is/are correct ?

$| \vec{p} | = | \vec{a} \times \vec{b} |$

$\vec{p} + \vec{q} + \vec{r} = 0$

$[\vec{p} \vec{q} \vec{r}] = 0$

$[\vec{a} \vec{b} \vec{c}] = 0$

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$\vec{p} = \vec{a} \times (\vec{a} \times \vec{b})$

$\Rightarrow | \vec{p} | = | \vec{a} \times \vec{b} |$

$\Rightarrow \vec{a}, \vec{b}, \vec{c}$ are coplanar

$\Rightarrow \vec{p}, \vec{q}, \vec{r}$ are coplanar