Question

⠀⠀
Let $\sf{\vec{r},\vec{a},\vec{b}$ and $\sf{\vec{c}}$ be four non-zero vectors such that $\vec{r}\cdotp\vec{a}$ = 0, $\sf{|\vec{r}\times\vec{b}|=|\vec{r}||\vec{b}|, |\vec{r}\times\vec{c}|=|\vec{r}||\vec{c}|}$, then find [a b c].

(1) |a| |b| |c|
(2) -|a| |b| |c|
(3) 0
(4) None​

Answer 1

Topic :-

Vectors

Given :-

$\overrightarrow{r},\overrightarrow{a},\overrightarrow{b}\:and\:\overrightarrow{c}\:are\:four\:non-zero\:vectors\:such\:that:-$

$\overrightarrow{r}\cdot\overrightarrow{a}=0$

$|\overrightarrow{r}\times\overrightarrow{b}|=|\overrightarrow{r} ||\overrightarrow{b} |$

$|\overrightarrow{r}\times\overrightarrow{c}|=|\overrightarrow{r} ||\overrightarrow{c} |$

To Find :-

$Value\:of\:[\overrightarrow{a}\:\:\overrightarrow{b}\:\:\overrightarrow{c}].$

Solution :-

$Let \:us\:assume\:that\:angle\:between\:\overrightarrow{r}\:and\:\overrightarrow{a}\:is\:\alpha,$

$angle\:between\:\overrightarrow{r}\:and\:\overrightarrow{b}\:is\:\beta\:and\:angle\:between\:\overrightarrow{r}\:and\:\overrightarrow{c}\:is\:\gamma.$

$\overrightarrow{r}\cdot\overrightarrow{a}=|\overrightarrow{r}||\overrightarrow{a}|\cos \alpha$

$0=|\overrightarrow{r}||\overrightarrow{a}|\cos \alpha$

$0=\cos \alpha$

$\boxed{\alpha=90^{\circ}}$

$|\overrightarrow{r}\times\overrightarrow{b}|=|\overrightarrow{r} ||\overrightarrow{b} |\sin\beta$

$|\overrightarrow{r} ||\overrightarrow{b} |=|\overrightarrow{r} ||\overrightarrow{b} |\sin\beta$

$1=\sin\beta$

$\boxed{\beta=90^{\circ}}$

$|\overrightarrow{r}\times\overrightarrow{c}|=|\overrightarrow{r} ||\overrightarrow{c} |\sin\gamma$

$|\overrightarrow{r} ||\overrightarrow{c} |=|\overrightarrow{r} ||\overrightarrow{c} |\sin\gamma$

$1=\sin\gamma$

$\boxed{\gamma=90^{\circ}}$

$\overrightarrow{a},\overrightarrow{b}\:and\:\overrightarrow{c}\:are\:perpendicular\:with\:\overrightarrow{r}\:which\:means$

$all\:three\:vectors\:will\:lie\:on\:same\:plane.$

$\because\:[\overrightarrow{x}\:\:\overrightarrow{y}\:\:\overrightarrow{z}]=0,if\:\overrightarrow{x},\overrightarrow{y}\:and\:\overrightarrow{z}\:are\:coplanar.$

$\therefore\:[\overrightarrow{a}\:\:\overrightarrow{b}\:\:\overrightarrow{c}]=0,as\:\overrightarrow{a},\overrightarrow{b}\:and\:\overrightarrow{c}\:are\:coplanar.$

Answer :-

$[\overrightarrow{a}\:\:\overrightarrow{b}\:\:\overrightarrow{c}]=0$

$Hence,\bold{option\:3}\:is\:correct\:option.$