Question Number 104035 by ajfour last updated on 19/Jul/20

Find the image of point OP^(→)  = p^�   in  the line  r^� =a^� +λb^�  .

Answered by mr W last updated on 19/Jul/20

Commented bymr W last updated on 19/Jul/20

AN=λb  (((p−a)∙b)/(∣b∣))=λ∣b∣  λ=(((p−a)∙b)/(∣b∣^2 ))  PN=a+λb−p  OQ=q=p+2PN=p+2(a+λb−p)  =2(a+λb)−p  ⇒q=2(a+(((p−a)∙b)/(∣b∣^2 ))b)−p

Commented byajfour last updated on 19/Jul/20

I believe this′ll be true for 3D even  Sir ?

Commented bymr W last updated on 19/Jul/20

basically yes.

Commented byajfour last updated on 19/Jul/20

((p+q)/2)=a+λb    ⇒   q=2(a+λb)−p  Now   (q−p).b=0         [(a+λb)−p].b = 0  λ=(((a−p).b)/(∣b∣^2 ))  q^� =(2a^� −p^� )+(((2(a^� −p^� ).b^� )/(∣b^� ∣^2 )))b^�

Commented bymr W last updated on 19/Jul/20

yes, thanks!