Question

please anyone answer all the questions as soon please

2 4 3 3 4 1. Given three points A = (0,–8, 10), B = (2, -5, 11), C = (-4,-9, 7) in R3. (a) Show that these three points are not collinear (not in a straight line). (b) Find the area of the triangle ABC. (c) Find the scalar equation of the plane containing the points A, B and C. (d) Find a point D on the plane such that ABCD is a parallelogram. What is the area of this parallelogram? 2. Let S be the plane 2x + 4y - 22 = -26 and T be the plane 3 + 3y - 3x = -18. If the planes intersect, give their line of intersection as a vector equation. 5 3. Given vectors i = (-9, -1, -6)", į = [10, 2, 7)" E R3. Determine whether the vector 6 = [7,-1, 4)" is in span{ū, ū}. If the vector is in the span then express it as a linear combination of ū, v. 4. Below are sets of vectors {ü, , w}. For each set determine whether the set is linearly independent or linearly dependent. If the set is linearly dependent, write one vector as a linear combination of the other two vectors. 2 (a) ü= -1 = w = 3 4 0 3 4 (b) ü= 1 -2 f

Answer 1

- 4. Given that: A-60,-8,10), B=(2, -5,41 ) =(-4,-9.7) of the Cross Product of the vector AB and he to zero then the points are not Co-llinean. is not equal Now AB=(2,3,1) Ã <-4,ad ,-3) now < ABX Ãe = RE 2 -4 -1 î [~9+1}-} [_6+43 + û [-2+123 =-8î t2 ŷ tloh a Hence Ło these three points are not collinear The area of ABC s the triangle £ 1 Abxael how | ABXÃ l = 5= 8) + (227+ (1032 64+ 4+100 (168 = 2542 how the area bei 2 x 2842 =542 let the equation of Plane be: aca-2014bc9-40) + C (2-20 1-0 where (lo, yo, zo) is a Point on Plane . {a,b,c) perpendicular to the plane.

V - AB =(2,3,1) V2 he =<-4,nd,-3) We have: ABX Ae -<-8,2,103 The equation of Plane bei -8 Ca-0)+269 +82 +10(2-10) = 0 8x +29+16 + 102-100 82-2y- 10Z + 84 = 0 OV 4a-y-5z +42-

(d) we have to find a Point D on the plane such that a parallelo sram ABCD is (0,-8,10)A D C2,4,2) B e (-4,-9,7) (2,-5, 11) Let D = (2,9,2) AS Diagonals of parallelo gram bisect each other. Вр o is the mid mid point of row well as AC the mid point of BD = the mid Point of Al - (242, 4= , 2+1)=(-470, -9-8 2 7410 2

so 042 을 닐 2 A= 2. y-5 날 y12 는 12 2 6 Henu, D =(-6, -12,6) AB 2,3,4) la md BE=<6,-4,-4) ~ 2 기 T 3 - 6 8 AB x B co - - C-1244 1 - 3 -94s] + A T- 8+lg ? - g + 28 - 100 area of the parallelo gram | AB x Bel a Nes)' (234 - IGE 2542 - N be! . 2.