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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 .

$$\mathrm{z}\:,\:\mathrm{z}_{\mathrm{1}} ,\:\mathrm{z}_{\mathrm{2}} \mathrm{being}\:\mathrm{on}\:\mathrm{the}\:\mathrm{same}\:\mathrm{line}, \\ $$$$\mathrm{z}−\mathrm{z}_{\mathrm{2}} \:\mathrm{is}\:\mathrm{parallel}\:\mathrm{to}\:\mathrm{z}_{\mathrm{1}} −\mathrm{z}_{\mathrm{2}} \:. \\ $$

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