If r . a = r . b = r . c =0 for some non-zero vector r , then the… - Vidyadip

MCQ

If $\vec{\text{r}}.\vec{\text{a}}=\vec{\text{r}}.\vec{\text{b}}=\vec{\text{r}}.\vec{\text{c}}=0$ for some non$-$zero vector $\vec{\text{r}},$ then the value of $\big[\vec{\text{a}}\vec{\text{ b }}\vec{\text{c}}\big],$ is:

A
$2$
B
$3$
✓
$0$
D
None of these

✓

Answer

Correct option: C.

$0$

If $\vec{\text{r}}.\vec{\text{a}}=0$ for some non$-$zero vector $\vec{\text{r}},$
then either $\vec{\text{a}}$ is a zero $-$ vector or it is perpendicular to $\vec{\text{r}}.$
If one of $\vec{\text{a}},\vec{\text{b}},\vec{\text{c}}$ is zero, then $\big[\vec{\text{a}}\vec{\text{b}}\vec{\text{c}}\big]=0$
If all $\vec{\text{a}},\vec{\text{b}}$ and $\vec{\text{c}}$ are non $-$ zero,
then they must be coplanar as they are perpendicular to vector $\vec{\text{r}}.$
$\therefore\big[\vec{\text{a}}\vec{\text{b}}\vec{\text{c}}\big]=0$

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