Let $\overrightarrow a = \widehat i - \alpha \widehat j + \beta \widehat k$, $\overrightarrow b = 3\widehat i + \beta \widehat j - \alpha \widehat k$ and $\overrightarrow c = -\alpha \widehat i - 2\widehat j + \widehat k$, where $\alpha$ and $\beta$ are integers. If $\overrightarrow a \,.\,\overrightarrow b = - 1$ and $\overrightarrow b \,.\,\overrightarrow c = 10$, then $$\left( {\overrightarrow a \, \times \overrightarrow b } \right).\,\overrightarrow c $$ is equal to ___________.
Answer (integer)
9
Solution
$\overrightarrow a = (1, - \alpha ,\beta )$<br><br>$\overrightarrow b = (3,\beta , - \alpha )$<br><br>$\overrightarrow c = ( - \alpha , - 2,1);\alpha ,\beta \in I$<br><br>$$\overrightarrow a \,.\,\overrightarrow b = - 1 \Rightarrow 3 - \alpha \beta - \alpha \beta = - 1$$<br><br>$\Rightarrow \alpha \beta = 2$
<br><br>Possible value of <br>$\alpha$ and $\beta$
<br><br>$$\matrix{
1 & 2 \cr
2 & 1 \cr
{ - 1} & { - 2} \cr
{ - 2} & { - 1} \cr
} $$<br><br>$\overrightarrow b \,.\,\overrightarrow c = 10$<br><br>$\Rightarrow - 3\alpha - 2\beta - \alpha = 10$<br><br>$\Rightarrow 2\alpha + \beta + 5 = 0$<br><br>$\therefore$ $\alpha$ = $-$2; $\beta$ = $-$1<br><br>$$[\overrightarrow a \,\overrightarrow b \,\overrightarrow c ] = \left| {\matrix{
1 & 2 & { - 1} \cr
3 & { - 1} & 2 \cr
2 & { - 2} & 1 \cr
} } \right|$$<br><br>$= 1( - 1 + 4) - 2(3 - 4) - 1( - 6 + 2)$<br><br>$= 3 + 2 + 4 = 9$
About this question
Subject: Mathematics · Chapter: Vector Algebra · Topic: Types of Vectors
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