# Need help on a vector question!

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Hi,

I'm stuck on a vector (I think?) question and need help.

Thank you so much in advance!

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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.

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.

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• 3 weeks later...

kw0573 is spot on

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