Three Dimensional Geometry Question 257
Question: Under what condition are the two lines $ y=\frac{m}{\ell }x+\alpha ,z=\frac{n}{\ell }x+\beta ; $ and $ y=\frac{m’}{\ell ‘}x+\alpha ‘,z=\frac{n’}{\ell ‘}x+\beta ’ $ Orthogonal?
Options:
A) $ \alpha \alpha ‘+\beta \beta ‘+1=0 $
B) $ (\alpha +\alpha ‘)+(\beta +\beta ‘)=0 $
C) $ \ell \ell ‘+mm’+nn’=1 $
D) $ \ell \ell ‘+mm’+nn’=0 $
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Answer:
Correct Answer: D
Solution:
[d] Given two lines are: $ y=\frac{mx}{\ell }+\alpha ,z=\frac{n}{\ell }x+\beta $ and $ y=\frac{m’}{\ell ‘}x+\alpha ‘,z=\frac{n’}{\ell ‘}x+\beta ’ $ These two lines can be represented as: $ \frac{y-\alpha }{m/\ell }=\frac{x-0}{1}=\frac{z-\beta }{n/\ell } $ And $ \frac{y-\alpha ‘}{m’/c’}=\frac{x-0}{1}=\frac{z-\beta ‘}{n’/\ell ‘} $ They are orthogonal, if $ \frac{m}{\ell }\times \frac{m’}{\ell ‘}+1\times 1+\frac{n}{\ell }\frac{n’}{\ell ‘}=-1\Rightarrow \ell \ell ‘+mm’+nn’=0 $
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