MHT CET · Maths · Vector Algebra
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.
- A \(\bar{a}, \bar{b}\) and \(\bar{c}\) are non-coplanar.
- B \(\bar{a}, \bar{b}\) and \(\bar{c}\) are coplanar.
- C \(\bar{a}, \bar{b}, \bar{c}\) are mutually perpendicular.
- D \(\bar{a}\) and \(\bar{b}\) are collinear.
Answer & Solution
Correct Answer
(A) \(\bar{a}, \bar{b}\) and \(\bar{c}\) are non-coplanar.
Step-by-step Solution
Detailed explanation
(C)
\(\bar{a}=3 \hat{i}+\hat{j}-\hat{k}, \bar{b}=2 \hat{i}-\hat{j}+7 \hat{k}, \bar{c}=7 \hat{i}-\hat{j}+23 \hat{k}\)
\(\begin{aligned}\left[\begin{array}{ccc}\bar{a} & \bar{b} & \bar{c}\end{array}\right] &=\left|\begin{array}{ccc}3 & 1 & -1 \\ 2 & -1 & 7 \\ 7 & -1 & 23\end{array}\right| \\ &=3(-23+7)-1(46-49)-1(-2+7) \\ &=3(-16)-(-3)-(5)=-50 \neq 0 \end{aligned}\)
\(\therefore \overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) are non coplanar.
\(\bar{a}=3 \hat{i}+\hat{j}-\hat{k}, \bar{b}=2 \hat{i}-\hat{j}+7 \hat{k}, \bar{c}=7 \hat{i}-\hat{j}+23 \hat{k}\)
\(\begin{aligned}\left[\begin{array}{ccc}\bar{a} & \bar{b} & \bar{c}\end{array}\right] &=\left|\begin{array}{ccc}3 & 1 & -1 \\ 2 & -1 & 7 \\ 7 & -1 & 23\end{array}\right| \\ &=3(-23+7)-1(46-49)-1(-2+7) \\ &=3(-16)-(-3)-(5)=-50 \neq 0 \end{aligned}\)
\(\therefore \overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) are non coplanar.
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