Menu Close

Prove-that-the-equation-of-the-line-joining-the-points-z-1-and-z-2-can-be-put-in-the-form-z-tz-1-1-t-z-2-where-t-is-a-parameter-




Question Number 19506 by Tinkutara last updated on 12/Aug/17
Prove that the equation of the line  joining the points z_1  and z_2  can be put  in the form z = tz_1  + (1 − t)z_2 , where  t is a parameter.
$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{line} \\ $$$$\mathrm{joining}\:\mathrm{the}\:\mathrm{points}\:{z}_{\mathrm{1}} \:\mathrm{and}\:{z}_{\mathrm{2}} \:\mathrm{can}\:\mathrm{be}\:\mathrm{put} \\ $$$$\mathrm{in}\:\mathrm{the}\:\mathrm{form}\:{z}\:=\:{tz}_{\mathrm{1}} \:+\:\left(\mathrm{1}\:−\:{t}\right){z}_{\mathrm{2}} ,\:\mathrm{where} \\ $$$${t}\:\mathrm{is}\:\mathrm{a}\:\mathrm{parameter}. \\ $$
Answered by ajfour last updated on 12/Aug/17
z−z_2 =t(z_1 −z_2 )  ⇒  z= tz_1 +(1−t)z_2  .
$$\mathrm{z}−\mathrm{z}_{\mathrm{2}} =\mathrm{t}\left(\mathrm{z}_{\mathrm{1}} −\mathrm{z}_{\mathrm{2}} \right) \\ $$$$\Rightarrow\:\:\mathrm{z}=\:\mathrm{tz}_{\mathrm{1}} +\left(\mathrm{1}−\mathrm{t}\right)\mathrm{z}_{\mathrm{2}} \:. \\ $$
Commented by Tinkutara last updated on 12/Aug/17
Thank you very much Sir!
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{Sir}! \\ $$
Commented by ajfour last updated on 12/Aug/17
z , z_1 , z_2 being on the same line,  z−z_2  is parallel to z_1 −z_2  .
$$\mathrm{z}\:,\:\mathrm{z}_{\mathrm{1}} ,\:\mathrm{z}_{\mathrm{2}} \mathrm{being}\:\mathrm{on}\:\mathrm{the}\:\mathrm{same}\:\mathrm{line}, \\ $$$$\mathrm{z}−\mathrm{z}_{\mathrm{2}} \:\mathrm{is}\:\mathrm{parallel}\:\mathrm{to}\:\mathrm{z}_{\mathrm{1}} −\mathrm{z}_{\mathrm{2}} \:. \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *