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  •   Let <img alt="\vec{r},\vec{a},\vec{b}\; and \; \vec{c}" src="https://learn.careers360.com/latex-image/?%5Cvec%7Br%7D%2C%5Cvec%7Ba%7D%2C%5Cvec%7Bb%7D%5C%3B%20and%20%5C%3B%20%5Cvec%7

 

Let \vec{r},\vec{a},\vec{b}\; and \; \vec{c}be four non - zero vectors  such that \vec{r}.\vec{a}=0,\left |\vec{r} \times \vec{b} \right |=\left | \vec{r} \right |\left | \vec{b} \right |, \left | \vec{r} \times \vec{c} \right |=\left | \vec{r} \right | \left | \vec{c} \right |,   then det\left ( a \; b \; c \right )=

Option: 1

\left | \vec{a} \right |\left | \vec{b} \right |\left | \vec{c} \right |


Option: 2

-\left | \vec{a} \right |\left | \vec{b} \right |\left | \vec{c} \right |


Option: 3

0


Option: 4

none of these


Answers (1)

best_answer

As we learned

Vector or cross product -

 \vec A\times \vec B= A\, B\sin \Theta

 And  Scalar , Dot or Inner Product -

 \vec{A}\cdot \vec{B}= A\, B\cdot \cos \Theta

 As given 

 \vec{r}.\vec{a}=0,\left |\vec{r} \times \vec{b} \right |=\left | \vec{r} \right |\left | \vec{b} \right |, \left | \vec{r} \times \vec{c} \right |=\left | \vec{r} \right | \left | \vec{c} \right |, 

\Rightarrow \vec{r}\perp \vec{a},\vec{b},\vec{c}

\therefore \vec{a},\vec{b},\vec{c} are coplanar

 then  determinant of  \vec{a},\vec{b},\vec{c} is zero .

Posted by

Rishi

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