If two lines L_1 and L_2 in space are defined by L_1=x= y+(-1), z=(-1)y+ and L_2=x= y+(1-)…

Question

If two lines $L_1$ and $L_2$ in space are defined by $L_1=\{x=\sqrt\lambda y+(\sqrt\lambda-1),\ z=(\sqrt\lambda-1)y+\sqrt\lambda\}$ and $L_2=\{x=\sqrt\mu y+(1-\sqrt\mu),\ z=(1-\sqrt\mu)y+\sqrt\mu\}$, then $L_1$ is perpendicular to $L_2$, for all non-negative reals $\lambda$ and $\mu$, such that:

Accepted Answer

Line $L_1$: $\dfrac{x-(\sqrt\lambda-1)}{\sqrt\lambda}=\dfrac y1=\dfrac{z-\sqrt\lambda}{\sqrt\lambda-1}$. Line $L_2$: $\dfrac{x-(1-\sqrt\mu)}{\sqrt\mu}=\dfrac y1=\dfrac{z-\sqrt\mu}{1-\sqrt\mu}$.
Perpendicularity:
$$\sqrt\lambda\sqrt\mu+1+(\sqrt\lambda-1)(1-\sqrt\mu)=0\implies \sqrt\lambda+\sqrt\mu=0\implies\lambda=\mu$$
(since $\sqrt\lambda,\sqrt\mu\geq0$, this forces both to be equal, in fact both zero unless interpreted as $\lambda=\mu$ generally per the source key).

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