Question

28 Consider (O:OAOB) an orthonormal system in space. Let G be the center of gravity of triangle ABC. 1° Calculate the coordinates of G 2°Consider the points A' (2 ;0:0) ,B, (0:2:0) and C" (0:0,3). a) Verify that these three points define a plane. b) Write a system of parametric equations of the plane (A'BC'). 3 Write a system of parametric equations of line (AC). 4° Verify that K (4:0-3) is the trace of the line (AC) with the plane (A'B'C'). 5° Verify that line (BC) cuts the plane (A'B'C') at L (0:4;-3). 6" a) Verify that (AB), (AB') and (KL) are parallel b) Determine the intersection (d) of the two planes (ABC) and (A'B'C'). Write a system of parametric equations of (d). Find the result of a).

Answer 1

in space Let Gl be the center or lavty of △Aec .Ga (a;0,0), Со;b:o), (0:0;e) osze. respectively Then the Co-arcknateof centre. fo,o,) to otbte ot 3 3 .3 abc la) These three PoinA delinea-plone.得队 The vectort Ae i(-2i2 0) Then the plane e with the. point A uthn ve have coein't halily the. plane (b) The ytem of faramehrc euation f the Plone A'bC' of the forcm 2, o: 0) (C0:0,3) (: 0,0)) t

(a-as-at j 26 3t) t.se R3] {[1WX) :: Ca,b.c) t เงิ t sa", Co,0,c) (0, a, o-o, c-o) V = (a,o,b) C-a ,o,c) The euat 3ven Ca,0,0) t t C-a,o, c) (x,y,ス) ニ Thus the, Parcamelt iC NHem of AC line スーtc