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Three Dimensional Geometry — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsThree Dimensional Geometry1 Mark
Question
State True or False for the following:
The line $\vec{\text{r}}=2\hat{\text{i}}-3\hat{\text{j}}-\hat{\text{k}}+\lambda({\text{i}}-{\text{j}}+2{\text{k}})$ lies in the plane $\vec{\text{r}}\cdot(3\hat{\text{i}}+{\text{j}}-{\text{k}})+2=0$
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Answer

False:Solution:
We have, $\vec{\text{r}}=2\hat{\text{i}}-3\hat{\text{j}}-\hat{\text{k}}+\lambda({\text{i}}-{\text{j}}+2{\text{k}})$
$\Rightarrow(\text{x}\hat{\text{i}}+\text{y}\hat{\text{j}}+\text{z}\hat{\text{k}})=(2+\lambda)\hat{\text{i}}+(-3-\lambda)\hat{\text{j}}+(-1+2\lambda)\hat{\text{k}}$
Position vector of any point on this line is
$(2+\lambda)\hat{\text{i}}+(-3-\lambda)\hat{\text{j}}+(-1+2\lambda)\hat{\text{k}}$
if this point lies on the then L.H.S of the plane is
$-6+3\lambda-3-\lambda+1-2\lambda+2$
$\neq0$
So, the line does not lie on the plane.

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