Find the distance between the skew lines r_(1)(t) =(1; 1;0) + (1, 6,2) t and r_(2)(s) =(1, 5, 2) +(2, 15, 6)s.

Find a nonzero vector orthogonal to the plane through the points P, Q, and R. and area of the triangle PQR
Consider the points below
P(1,0,1) , Q(-2,1,4) , R(7,2,7)
a) Find a nonzero vector orthogonal to the plane through the points P,Q and R
b) Find the area of the triangle PQR

Find the volume of the parallelepiped with one vertex at the origin and adjacent vertices at ${\displaystyle (1,3,0),(-2,0,2),\text{and}(-1,3,-1)}$.

Find the vectors T, N, and B at the given point.
$r(t)=<t^2,\frac{2}{3}{t}^3,t>$ and point $<4,-\frac{16}{3},-2>$

If A is an n x n matrix , where are the entries on the main diagonal of A-A^T? Justify yoyr answer.

We have $F(x,y)\in \mathbb{Z}[X,Y]$ a positive definite binary form of degree $\ge 3$. I have to prove, without using lower bounds on linear forms in logarithms (we were working with Baker's theorems), that for each positive integer m the equation $F(x,y)=m$ has only finitely many solutions in $x,y\in \mathbb{Z}$. I also have to describe a method to find these.
We know that the coefficient of $X^d$ of $F$ is positive, and that all zeros of $F(X,1)$ are in $\mathbb{C}\mathrm{\setminus }\mathbb{R}$. We also know the discriminant of $F$ is negative.
I have trouble finding this out by myself. I also looked up some number theory books, but most literature is about binary quadratic forms, not the binary form I have to prove this for. Hope someone can help me!

Proove that the set of oil $2\times 2$ matrices with entries from R and determinant +1 is a group under multiplication

1. Show that sup ${\displaystyle \{1-\frac{1}{n}:n\in N\}=1\{1-\frac{1}{n}:n\in N\}=\frac{1}{2}}$. If ${\displaystyle S=\{\frac{1}{n}-\frac{1}{m}:n,m\in N\}S=\{\frac{1}{n}-\frac{1}{m}:n,m\in N\},f\in d\in fS\text{and}\supset S.end\left\{tab\underset{―}{a}r\right\}}$