Let the volume of a parallelopiped whose
coterminous edges are given by
$\overrightarrow u = \widehat i + \widehat j + \lambda \widehat k$, $\overrightarrow v = \widehat i + \widehat j + 3\widehat k$ and
$\overrightarrow w = 2\widehat i + \widehat j + \widehat k$ be 1 cu. unit. If $\theta$ be the angle between the
edges $\overrightarrow u$ and $\overrightarrow w$ , then cos$\theta$ can be :
Solution
Volume of parallelopiped = 1
<br><br>$$\left| {\left[ {\matrix{
{\overrightarrow u } & {\overrightarrow v } & {\overrightarrow w } \cr
} } \right]} \right|$$ = 1
<br><br>$\Rightarrow$ $$\left| {\matrix{
1 & 1 & \lambda \cr
1 & 1 & 3 \cr
2 & 1 & 1 \cr
} } \right|$$ = $\pm$1
<br><br>$\Rightarrow$ $\lambda$ = 2, 4
<br><br>$\overrightarrow u = \widehat i + \widehat j + 2 \widehat k$ or
<br>$\overrightarrow u = \widehat i + \widehat j + 4 \widehat k$
<br><br>$\therefore$ cos $\theta$ = $${{{\overrightarrow u .\overrightarrow w } \over {\left| {\overrightarrow u } \right|\left| {\overrightarrow w } \right|}}}$$
<br><br>= ${{2 + 1 + 4} \over {\sqrt {18} \sqrt 6 }}$ or ${{2 + 1 + 2} \over {\sqrt 6 \sqrt 6 }}$
<br><br>= ${7 \over {6\sqrt 3 }}$ or ${5 \over 6}$
About this question
Subject: Mathematics · Chapter: Vector Algebra · Topic: Types of Vectors
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