Let $\overrightarrow c$ be a vector perpendicular to the vectors, $\overrightarrow a$ = $\widehat i$ + $\widehat j$ $-$ $\widehat k$ and
$\overrightarrow b$ = $\widehat i$ + 2$\widehat j$ + $\widehat k$. If $\overrightarrow c \,.\,\left( {\widehat i + \widehat j + 3\widehat k} \right)$ = 8 then the value of
$\overrightarrow c$ . $\left( {\overrightarrow a \times \overrightarrow b } \right)$ is equal to __________.
Answer (integer)
28
Solution
$$\overrightarrow a \times \overrightarrow b = \left| {\matrix{
{\widehat i} & {\widehat j} & {\widehat k} \cr
1 & 1 & { - 1} \cr
1 & 2 & 1 \cr
} } \right| = (3, - 2,1)$$<br><br>$$\overrightarrow c \bot \overrightarrow a ,\overrightarrow c \bot \overrightarrow b \Rightarrow C||\overrightarrow a \times \overrightarrow b $$<br><br>$\overrightarrow c = \lambda (\overrightarrow a \times \overrightarrow b )$<br><br>$$ \Rightarrow \overrightarrow c = \lambda (3\widehat i - 2\widehat j + \widehat k)$$<br><br>Given, $\overrightarrow c .(\widehat i + \widehat j + 3\widehat k) = 8$<br><br>$\Rightarrow 3\lambda - 2\lambda + 3\lambda = 8$<br><br>$\Rightarrow 4\lambda = 8 \Rightarrow \lambda = 2$<br><br>$\therefore$ $\overrightarrow c = 6\widehat i - 4\widehat j + 2\widehat k$<br><br>$$\overrightarrow c \,.\,(\overrightarrow a \times \overrightarrow b ) = [\overrightarrow c \overrightarrow a \overrightarrow b ] = \left| {\matrix{
6 & { - 4} & 2 \cr
1 & 1 & { - 1} \cr
1 & 2 & 1 \cr
} } \right|$$<br><br>$\Rightarrow$ 18 + 8 + 2 = 28
About this question
Subject: Mathematics · Chapter: Vector Algebra · Topic: Types of Vectors
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