Ok, keep in mind that the general equation of a plane is Ax+By+Cz+D=0 .
It can be proven that the vector \bf{N}=(A,B,C) , called normal vector, is orthogonal to the plane Ax+By+Cz+D=0 . So two parallel planes have a common normal vector \bf{N}.
What do you need to be able to talk about the...
Homework Statement
Let [G,+,0] be a non-abelian group with a binary operation + and a zero element 0 .
To prove that if both the zero element and the inverse element act on the same side, then they both act the other way around, that is:
If \forall a \in G ,
a + 0 = a ,
and
a + (-a)...
The union of two probability sets is the probability of A happening OR B happening:
The chances of throwing a dice and getting 6 is 1/6, the chance of getting a 2 is 1/6, the chance of getting a 6 or a 2 is 1/6 + 1/6 = 1/3.
The intersection of two probability sets, which is the probability of A...
Um, I might be wrong here, long since my last prob class but I believe you're almost there:
For ijk +mn to be even IJK AND MN must be either odd or even, so why not calculate the probability of BOTH being odd (intersection between the chances of ikj being odd and mn being odd), and the...
It is not very elegant but the proof follows from the principles of order on the real number field.
Considering positive primes: given p_{1} < p_{2}, then it follows that p_{1} < p_{1}^2 < p_{1} \cdot p_{2}, by inductive reasoning you can prove that p_{n+1}<p_{n+1} \cdot p_{1} < p_{n+1} \cdot...
Huge hint:
In the main diagonal of your matrix i = j, so max{ i, j } = i or j
above the main diagonal j > i, so max{ i, j } = j
below the main diagonal i > j so max { i, j } = i
Homework Statement
Let A, B be both matrices with the same dimensions. Is AB^2 = (A^2)(B^2) a valid claim?
Homework Equations
The Attempt at a Solution
I attempted to show that (AB)^2 = (AB)(AB) = A(BA)B
and that (A^2)(B^2) = (AA)(BB) = A(AB)B, so for A(BA)B to be equal to A(AB)B, AB...
Like Mark44 said, use the fact that A - B = A + (-B).
Hint: Sine is an odd function, which means that f(-x) = -f(x)
2nd hint: Cosine is an even function, which means that f(-x) = f(x)
That should do the trick.