Question

If the vectors $2i-3j+4k$ and $i+2j-k$ and $mi-j+2k$ are coplanar, then the value of $m$ is?

• A. $\frac{5}{8}$
• B. $\frac{8}{5}$
• C. $-\frac{7}{4}$
• D. $\frac{2}{3}$

Answer 1

The correct option is B

$\frac{8}{5}$

Explanation for the correct option :

Step 1. Find the value of $m$.

Given three vectors:

$$\begin{gathered} a=2i-3j+4k \\ b=i+2j-k \\ c=mi-j+2k \end{gathered}$$

As the three vectors $a,b,c$ are coplanar so the condition is $\left[a,b,c\right]=0$. Thus:

$\begin{array}{rcl}\left|\begin{array}{ccc}2& -3& 4\\ 1& 2& -1\\ m& -1& 2\end{array}\right|& =& 0\\ 2\left[2\times 2-\left(-1\right)\times \left(-1\right)\right]-\left(-3\right)\left[1\times 2-\left(-1\right)\times m\right]+4\left[1\times \left(-1\right)-2\times m\right]& =& 0\\ 2\left[4-1\right]+3\left[2+m\right]+4\left[-1-2m\right]& =& 0\\ 6+6+3m-4-8m& =& 0\\ -5m+8& =& 0\\ -5m& =& -8\\ m& =& \frac{8}{5}\end{array}$

Hence, the correct option is B.