the points a b c d are concordantly ( 1,2,-3) , (-1,2,1) , ( 0,1,-2) , ( 2,-1,1)
find formula of the plane going thorugh d and which is pararlel to plane abc
calculate the volume of pyramid abcd.
attemp to solve:
the vectors ab ac ad are concordantly -204 -1-11 1-34
using the determinant: $ \left| \begin{array}{ccc} \ x-1& y-2 & z+3 \\ -2 & \ 0 & 4 \\ -1 & -1 & \ 1 \end{array} \right| $ whihc yields: 2x+y +z-1=0 - formula for plane abc
normal is hence 2 1 1 and combining that with the info on point d we get that d (free argument) for desired plane is -4 and therfore its formula is 2x+y+z-4=0
for the volume we calulate that $ab\cdot ac$ (dot product) is 6, and the scalar product of their sizes is sqrt of 60. arccos of ratio of the two is 39.23 degrees. the volume would be sixth of the scalr prosuct of sizes of ab and ac and sin of said angle (39.23), yielding 0.816. I got other results in the textbook so I would like to ask help on this one.
Assuming your determinant calculation is correct, your work for the plane looks fine to me. I'm not seeing how to do the pyramid problem in my mind, so I can't check that for you.
As a side note: the dot product ($\cdot$) is the scalar product. Thus, I don't really understand what you're saying in your last paragraph ("the dot product is 6, and the scalar product is...") The scalar product and dot product are the same thing.