MHT CET · Maths · Three Dimensional Geometry
If \(\bar{a}=3 \hat{i}+\hat{j}-\hat{k}, \bar{b}=2 \hat{i}-\hat{j}+23 \hat{k}\) and \(\bar{c}=7 \hat{i}-\hat{j}+23 \hat{k}\), then which of the following is valid.
- A \(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) are mutually perpendicular
- B \(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) are non-coplanar
- C \(\overline{\mathrm{a}}\) and \(\overline{\mathrm{b}}\) are collinear
- D \(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) are coplanar
Answer & Solution
Correct Answer
(B) \(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) are non-coplanar
Step-by-step Solution
Detailed explanation
We have \(\bar{a}=3 \hat{i}+\hat{j}-\hat{k}, \bar{b}=2 \hat{i}-\hat{j}+23 \hat{k}\) and \(\bar{c}=7 \hat{i}-\hat{j}+23 \hat{k}\) \(\bar{a} \cdot \bar{b}=6-1-23 \neq 0\) and \(\bar{b} \cdot \bar{c}=14+1+529 \neq 0\)
Thus \(\bar{a}, \bar{b}, \bar{c}\) are non mutually perpendicular.
Also for \(\overline{\mathrm{a}}\) and \(\overline{\mathrm{b}}, \frac{3}{2} \neq-1 \neq \frac{-1}{23}\).
Thus \(\bar{a}\) and \(\bar{b}\) are not collinear.
Now \(\left|\begin{array}{ccc}3 & 1 & -1 \\ 2 & -1 & 23 \\ 7 & -1 & 23\end{array}\right|=3(-23+23)-(46-161)-(-2\) \(+7)\neq 0\).
Thus \(\bar{a}, \bar{b}, \bar{c}\) are non coplanar.
Thus \(\bar{a}, \bar{b}, \bar{c}\) are non mutually perpendicular.
Also for \(\overline{\mathrm{a}}\) and \(\overline{\mathrm{b}}, \frac{3}{2} \neq-1 \neq \frac{-1}{23}\).
Thus \(\bar{a}\) and \(\bar{b}\) are not collinear.
Now \(\left|\begin{array}{ccc}3 & 1 & -1 \\ 2 & -1 & 23 \\ 7 & -1 & 23\end{array}\right|=3(-23+23)-(46-161)-(-2\) \(+7)\neq 0\).
Thus \(\bar{a}, \bar{b}, \bar{c}\) are non coplanar.
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