Question

2) If the volume of a parallelopiped whose coterminous edges are -31 + 2 +nk, 2i+j-k -i+3j+2k is 7 cu-units, then the value of n is
(a) o
(b) 1
(c) 3
(d) 4




piz .. help mi​

Answer 1

$\large\underline{\sf{Solution-}}$

Given three coterminous edges of parallelopiped.

Let assume that three edges be represented as

$\rm \: \vec{a} = - 3\hat{i} + 2\hat{j} + n\hat{k}$

$\rm \: \vec{b} = 2\hat{i} + \hat{j} - \hat{k}$

$\rm \: \vec{c} = - \hat{i} + 3\hat{j} + 2\hat{k}$

Also, given that

$\rm \: Volume_{(parallelopiped)} = 7 \: cubic \: units$

We know, volume of parallelopiped is given by

$\rm \: Volume_{(parallelopiped)} = [\vec{a} \: \: \vec{b} \: \: \vec{c}]$

$\rm \:\bigg | \begin{gathered}\sf \left | \begin{array}{ccc} - 3&2&n\\2&1& - 1\\ - 1&3&2\end{array}\right | \end{gathered}\bigg | = 7$

$\rm \: | - 3(2 + 3) - 2(4 - 1) + n(6 + 1)| = 7$

$\rm \: | - 3(5) - 2(3) + n(7)| = 7$

$\rm \: | - 15 - 6 + 7n| = 7$

$\rm \: | - 21 + 7n| = 7$

$\rm \: 7n - 21 = \: \pm \: 7$

$\rm \: 7n - 21 = 7 \: \: or \: \: 7n - 21 = - 7$

$\rm \: 7n = 28 \: \: or \: \: 7n = 14$

$\rm\implies \:n = 4 \: \: or \: \: n = 2$

So, option (d) is correct.

$\rule{190pt}{2pt}$

Additional Information :-

$\boxed{ \rm{ \:[\vec{a} \: \: \vec{b} \: \: \vec{c}] \: = \: \vec{a}.(\vec{b} \times \vec{c}) \: }}$

$\boxed{ \rm{ \:[\vec{a} \: \vec{b} \: \vec{c}] \: = \: [\vec{b} \: \vec{c} \: \vec{a}] \: = \: [\vec{c} \: \vec{a} \: \vec{b}] \: }}$

$\boxed{ \rm{ \:[\vec{a} \: \: \vec{a} \: \: \vec{b}] \: = \: 0 \: }}$

$\boxed{ \rm{ \:[\hat{i} \: \: \hat{j} \: \: \hat{k}] \: = \: [\hat{j} \: \: \hat{k} \: \: \hat{i}] \: = \: [\hat{k} \: \: \hat{i} \: \: \hat{j}] \: = \: 1 \: }}$

$\boxed{ \rm{ \:[m \: \vec{a} \: \: \vec{b} \: \: \vec{c}] \: = \: m \: [\vec{a} \: \: \vec{b} \: \: \vec{c}] \: }}$

$\boxed{ \rm{ \:[\vec{a} \: \: \vec{b} \: \: \vec{c}] \: = \: 0 \: \rm\implies \:\vec{a}, \: \vec{b}, \: \vec{c} \: are \: coplanar \: }}$