If $\bar{a}=3 \hat{\imath}+\hat{\jmath}-\hat{k}, \bar{b}=2 \hat{\imath}-\hat{\jmath}+7 \hat{k}$ and…

If $\bar{a}=3 \hat{\imath}+\hat{\jmath}-\hat{k}, \bar{b}=2 \hat{\imath}-\hat{\jmath}+7 \hat{k}$ and $\bar{c}=7 \hat{\imath}-\hat{\jmath}+23 \hat{k}$ are three vectors, then which of the following statement is true.
  1. $\bar{a}, \bar{b}$ and $\bar{c}$ are non-coplanar.
  2. $\bar{a}, \bar{b}$ and $\bar{c}$ are coplanar.
  3. $\bar{a}, \bar{b}, \bar{c}$ are mutually perpendicular.
  4. $\bar{a}$ and $\bar{b}$ are collinear.

Solution

$\bar{a}=3 \hat{i}+\hat{j}-\hat{k}, \bar{b}=2 \hat{i}-\hat{j}+7 \hat{k}, \bar{c}=7 \hat{i}-\hat{j}+23 \hat{k}$ $\begin{aligned}\left[\begin{array}{ccc}\bar{a} & \bar{b} & \bar{c}\end{array}\right] &=\left|\begin{array}{ccc}3 & 1 & -1 \\ 2 & -1 & 7 \\ 7 & -1 & 23\end{array}\right| \\ &=3(-23+7)-1(46-49)-1(-2+7) \\ &=3(-16)-(-3)-(5)=-50 \neq 0 \end{aligned}$ $\therefore \overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}$ are non coplanar.

Asked in: MHT CET 2020 (19 Oct Shift 2)

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