ebu 74 Posted January 28, 2016 Report Share Posted January 28, 2016 Hi, I'm stuck on a vector (I think?) question and need help. Thank you so much in advance! Reply Link to post Share on other sites

kw0573 1,252 Posted January 28, 2016 Report Share Posted January 28, 2016 Hi There are 7 cases where 3 planes can intersect. Trivial cases: 1) three planes identical, plane of solutions 2) two planes identical third plane different normal: line of sol'n 3) two planes identical third plane is parallel, no solution 4) all three planes parallel non intersecting, also no solution Non-trivial cases Where the triple scalar product of the 3 normals = 0 (indicating the 3 distinct normals are coplanar and each is linearly dependent of the other two 5) If infinite solution, the line intersection of 2 planes is also on the third plane (tsp and this are two equations). 6) If no solution, the line intersection of 2 planes does not lie on the third. Where tsp =/= 0, the three normals form a basis for a 3-d space 7) Use tsp to solve for alpha. Plug the line intersection of known equations to solve for beta. Alternatively, if your school taught you about reduced echelon form, for 5) you get a row of all 0s in the augmented matrix 6) you get inconsistency (0 = non-zero number) 7) you get variable = value (eg z = 4) I got a) i) alpha = 2, beta =/= 0 ii) alpha =/= 2, beta is real (any real because plane 3 cannot be same or parallel to plane 1 or 2) iii) alpha = 2, beta = 0 b) EDIT: Initally I put vector form but I didn't read question correctly I solved it using RREF (reduced row echelon form) but if you are unable to solve it using triple scalar product and plug in the equation of a line let me know. EDIT: I was able to get the answer without using RREF or matrix operations. Note the point (-2, 4, 0) is the point you would get using RREF. If not you can use any other point on the line but anyhow (-2, -2, 1) is the direction vector. Reply Link to post Share on other sites

Jai Chandak 9 Posted February 16, 2016 Report Share Posted February 16, 2016 Here is the Mark scheme's answer to your question kw0573 is spot on Reply Link to post Share on other sites

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