If the volume of a parallelopiped, whose
coterminus edges are given by the
vectors $\overrightarrow a = \widehat i + \widehat j + n\widehat k$,
$\overrightarrow b = 2\widehat i + 4\widehat j - n\widehat k$ and
$\overrightarrow c = \widehat i + n\widehat j + 3\widehat k$ ($n \ge 0$), is 158 cu. units, then :
Solution
We know, Volume(V) = $\left[ {\overrightarrow a \overrightarrow b \overrightarrow c } \right]$
<br><br>$\Rightarrow$ 158 = $$\left| {\matrix{
1 & 1 & n \cr
2 & 4 & { - n} \cr
1 & n & 3 \cr
} } \right|$$
<br><br>$\Rightarrow$ (12 + n<sup>2</sup>) – (6 + n) + n(2n–4)=158
<br><br>$\Rightarrow$ 3n<sup>2</sup>
–5n + 6 –158 = 0
<br><br>$\Rightarrow$ 3n<sup>2</sup>
– 5n – 152 = 0
<br><br>$\Rightarrow$ 3n<sup>2</sup>
– 24n + 19n – 152 = 0
<br><br>$\Rightarrow$ (3n + 19) (n–8) = 0
<br><br>$\Rightarrow$ n = 8, $- {{19} \over 3}$ (rejected)
<br><br>$\therefore$ $\overrightarrow a = \widehat i + \widehat j + 8\widehat k$, <br><br>$\overrightarrow b = 2\widehat i + 4\widehat j - 8\widehat k$ and <br><br>$\overrightarrow c = \widehat i + 8\widehat j + 3\widehat k$
<br><br>Now ${\overrightarrow a .\overrightarrow c }$ = 1 + 8 + 24 = 33
<br><br>${\overrightarrow b .\overrightarrow c }$ = 2 + 32 - 24 = 10
About this question
Subject: Mathematics · Chapter: Vector Algebra · Topic: Types of Vectors
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