MHT CET · Maths · Three Dimensional Geometry
If the line \(\bar{r}=(\hat{\imath}-2 \hat{\jmath}+3 \hat{k})+\lambda(2 \hat{\imath}+\hat{\jmath}+2 \hat{k})\) is parallel to the plane \(\bar{r} .(3 \hat{\imath}-2 \hat{\jmath}+m \hat{k})=10\), then the value of \(\mathrm{m}\) is
- A 2
- B -3
- C -2
- D 3
Answer & Solution
Correct Answer
(C) -2
Step-by-step Solution
Detailed explanation
(B)
\(\overline{\mathrm{b}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+2 \hat{\mathrm{k}}\) and \(\overline{\mathrm{n}}=3 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\mathrm{m} \hat{\mathrm{k}}\)
Since, line is parallel to the plane, \(\bar{b} \cdot \bar{n}=0\)
\((\hat{2} \mathrm{i}+\hat{\mathrm{j}}+2 \hat{\mathrm{k}}) \cdot(3 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\mathrm{m} \hat{\mathrm{k}}) =0 \Rightarrow 6-2+2 \mathrm{~m}\) \(=0 \Rightarrow \mathrm{m}=-2\)
\(\overline{\mathrm{b}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+2 \hat{\mathrm{k}}\) and \(\overline{\mathrm{n}}=3 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\mathrm{m} \hat{\mathrm{k}}\)
Since, line is parallel to the plane, \(\bar{b} \cdot \bar{n}=0\)
\((\hat{2} \mathrm{i}+\hat{\mathrm{j}}+2 \hat{\mathrm{k}}) \cdot(3 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\mathrm{m} \hat{\mathrm{k}}) =0 \Rightarrow 6-2+2 \mathrm{~m}\) \(=0 \Rightarrow \mathrm{m}=-2\)
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