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Finding plane equation given two lines - Mathematics Stack Exchange Determine an equation of the plane containing the lines x−1 2 = y+1 −1 = z−5 6 x 1 2 = y + 1 1 = z 5 6; r =<1, −1, 5> +t <1, 1, −3> r =<1, 1, 5> + t <1, 1, 3> I calculated the cross product between the directional vector of both lines to find the normal vector n n, but when I looked for an intersection point r0 r 0 to apply the formula: <r −r0> ∙ n <r r 0> n, I did not find any
Summation of squared 3-j symbol - Mathematics Stack Exchange where Δ(j1,j2,j3) Δ (j 1, j 2, j 3) is the triangular condition i e if it is satisfied, it equals to the unit otherwise is zero I have checked this property numerically and it's working well
Prove that $\\sum_{j=0}^{n}\\sum_{i=j}^{n} {^nC_i} {^iC_j}=3^n$ 1 Prove that ∑j=0n ∑i=jn nCiiCj =3n ∑ j = 0 n ∑ i = j n n C i i C j = 3 n I have tried by interchanging j by i but unable to prove Don't know which combination formula would be used Please help
summation - Rules of Double Sums - Mathematics Stack Exchange Question related to (3): why suddenly an index j j appears (initially, we had only i i)? Furthermore, in relation to (3) there is a theorem given in my textbook without intuition proof: When we work with double sums, the following theorem is of special interest; it is an immediate consequence of the multinominal expansion of (x1 +x2 + … +xn)2 (x 1 + x 2 + + x n) 2: Theorem:
Binomial Theorem with Three Terms - Mathematics Stack Exchange For instance, suppose you want the coefficient of x15 x 15 for (x7 +x6 +x5 +x4 +x3)3 (x 7 + x 6 + x 5 + x 4 + x 3) 3 You then need to find the number of ways to take from [7,6,5,4,3] with replacement three times and get a sum of 15