Question

prove that the points (1,2,3), (4,0,4), (-2,4,2), (7,-2,5) are collinear.

Answer 1

IF THEIR area of triangle is 0 then they are collinear

Answer 2

Thus, The above points are col-linear.

Hence Proved.

Step-by-step explanation:

Given,

Four points are $(1,2,3),(4,0,4),(-2,4,2)$ and $(7,-2,5)$

$P$ point is $(1,2,3)$

$Q$ point is $(4,0,4)$

$R$ point is $(-2,4,2)$

$S$ point is $(7,-2,5)$

Distance between $PQ=\sqrt{(4-1)^{2} +(0-2)^{2} +(4-3)^{2} }$

                          ⇒$PQ=\sqrt{9+4+1} =\sqrt{14}$

Distance between $QR=\sqrt{(-2-4)^{2}+(4-0)^{2} +(2-4)^{2} }$

                          ⇒$QR=\sqrt{36+16+4} =2\sqrt{14}$

Distance between $RS=\sqrt{(7+2)^{2}+(-2-4)^{2} +(5-2)^{2} }$

                         ⇒$RS=\sqrt{81+36+9} =\sqrt{126} =3\sqrt{14}$

Distance between $PS=\sqrt{(7-1)^{2} +(-2-2)^{2} +(5-3)^{2} }$

                         ⇒$PS=\sqrt{36+16+4} =\sqrt{56} =2\sqrt{14}$

From above distance,

Distance between $(PQ+RS)=$Distance between $(QR+PS)$

⇒$(\sqrt{14}+3\sqrt{14} )=(2\sqrt{14}+ (2\sqrt{14})$

∴$4\sqrt{14}=4\sqrt{14}$

So, The above points are col-linear.

Hence Proved.