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Three Dimensional Geometry — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsThree Dimensional Geometry5 Marks
Question
Determine whether the following pair of lines intersect or not:
$\vec{\text{r}}=\big(\hat{\text{i}}-\hat{\text{j}}\big)+\lambda\big(2\hat{\text{i}}+\hat{\text{k}}\big)$ and $\vec{\text{r}}=\big(2\hat{\text{i}}-\hat{\text{j}}\big)+\mu\big(\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}\big)$
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Answer

Given equation of lines are
$\vec{\text{r}}=\big(\hat{\text{i}}-\hat{\text{j}}\big)+\lambda\big(2\hat{\text{i}}+\hat{\text{k}}\big)$ and $\vec{\text{r}}=\big(2\hat{\text{i}}-\hat{\text{j}}\big)+\mu\big(\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}\big)$
If these lines intersect each other, there must be some common point, So, we must have $\lambda$ and $\mu$ such that
$\big(\hat{\text{i}}-\hat{\text{j}}\big)+\lambda\big(2\hat{\text{i}}+\hat{\text{k}}\big)=\big(2\hat{\text{i}}-\hat{\text{j}}\big)+\mu\big(\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}\big)$
$(1+2\lambda)\hat{\text{i}}-\hat{\text{j}}+\lambda\hat{\text{k}}=(2+\mu)\hat{\text{i}}+(-1+\mu)\hat{\text{j}}-\mu\hat{\text{k}}$
Equation the cofficients of $\hat{\text{i}},\hat{\text{j}}$ and $\hat{\text{k}},$
$1+2\lambda=2+\mu\Rightarrow2\lambda-\mu=1\dots(1)$
$-1=-1+\mu\Rightarrow\mu=0\dots(2)$
$\lambda=-\mu\Rightarrow\lambda=0\dots(3)$
Put value of $\lambda$ and $\mu$ in equation (1),
$2\lambda=\mu=1$
$2(0)-(0)=1$
$0=1$
$\text{LHS}\neq\text{RHS}$
Since, the values of $\lambda$ and $\mu$ form equation (2) and (3) dose not satisfy equation (1),
Hence, given lines do not intersect each other.

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