Question

If the volume of a parallelopiped, whose coterminous edges are given by the vectors $a=i+j+nk,b=2i+4j-nk,$ and $c=i+nj+3k;\left(n\ge 0\right)$, is $158$ cubic units, then:

• A. $a·c=17$
• B. $b·c=10$
• C. $n=9$
• D. $n=7$

Answer 1

The correct option is B

$b·c=10$

Explanation for the correct option

Option B: $b·c=10$

Step 1. Find the value of $n$.

A parallelopiped is defined by the edges $a=i+j+nk,b=2i+4j-nk,$ $c=i+nj+3k;\left(n\ge 0\right)$ and its volume is $158$ cubic units, so $\left[abc\right]=158$. Thus:

$\begin{array}{rcl}\left|\begin{array}{ccc}1& 1& n\\ 2& 4& -n\\ 1& n& 3\end{array}\right|& =& 158\\ 1\left[12-\left(-n^2\right)\right]-1\left[6-\left(-n\right)\right]+n\left[2n-4\right]& =& 158\\ 12+n^2-6-n+2n^2-4n-158& =& 0\\ 3n^2-5n-152& =& 0\\ 3n^2-24n+19n-152& =& 0\\ 3n(n-8)+19(n-8)& =& 0\\ (n-8)(3n+19)& =& 0\end{array}$

Either

$\begin{array}{rcl}n-8& =& 0\\ n& =& 8\end{array}$ or

$\begin{array}{rcl}3n+19& =& 0\\ 3n& =& -19\\ n& =& -\frac{19}{3}\end{array}$

As $n$ cannot be negative so the value of $n$ is $8$.

Step 2. Find the value of $b·c$

For $n=8$ the vectors $b$ and $c$ are defined as: $b=2i+4j-8k;c=i+8j+3k$. And the dot product is given as;

$\begin{array}{rcl}b·c& =& (2i+4j-8k)·(i+8j+3k)\\ & =& 2·1+4·8+(-8)·3\\ & =& 2+32-24\\ & =& 10\end{array}$

Hence, the correct option is B.

Explanation for the incorrect options

Option A: $a·c=17$

Find the value of $a·c$

For $n=8$ the vectors $a$ and $c$ are defined as: $a=i+j+8k;c=i+8j+3k$. And the dot product is given as;

$\begin{array}{rcl}a·c& =& (1i+1j+8k)·(i+8j+3k)\\ & =& 1·1+1·8+8·3\\ & =& 1+8+24\\ & =& 33\end{array}$

As, $a·c\ne 17$

Hence, option A is incorrect.

Option C: $n=9$

We know that, $n=8$.

Hence, option A is incorrect.

Option D: $n=7$

We know that, $n=8$.

Hence, option A is incorrect.

Hence, option B is correct.