Let v1433 v212117 and y3631h For what value of h is y in

Let v1=[-4,3,-3] , v2=[-12,11,-7] , and y=[36,-31,h] . For what value of h is y in the plane spanned by v1 and v2? h=

Solution

y = c V1+ d V2 for some real c and d (say)

(36,-31,h) = c(-4,3,-3) +d(-12,11,-7)

=> (36,-31,h) =(-4c,3c,-3c)+(-12d,11d,-7d)

=> (36,-31,h) =(-4c-12d,3c+11d,-3c-7d)

=> -4c-12d=36, 3c+11d=-31, -3c-7d=h

=> c+ 3d = -9 , 3c+11d =-31, -3c-7d=h

=> 3c+ 9d =-27 , 3c+11d=-31, -3c -7d =h

Subtracting 2nd from 1st equation,we get

9d-11d =-27+31

-2d = 4

=> d = 4/-2 =-2

=> c= -9-3d = -9-3(-2)=-3

h= -3c-7d = -3(-3) +29*(-2)==9-58=-49(answer)