If the position vectors of points T and S are 3a, - 2a, +a, and 4a, + 6a, + 2a, respectively, find (a) the coordinates of T and S, (b) the distance vector from T to S, (c) the distance between T and S.

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

Let $\phi :K[x{]}_{\le n}\to K[x{]}_{\le n-1}$ with $\phi$ the linear transformation defind by $\phi (f)=f^{\prime }$. Select a base and find the matrix of the linear transformation.
I took the standard basis for grade-n polynomials:
$B=<1,x,x^2,\dots ,x^n>,\phi (B)=\phi (1,x,x^2,...,x^n)=(0,2x,...,n{x}^{n-1})$
So, the matrix is $\left[\begin{array}{c}0\\ 2x\\ \dots \\ n{x}^{n-1}\end{array}\right]$?

Find the equation of the line that satisfies the given conditions:
Goes through $(-1,4)$, slope $-9$

Find the transformation matrix
Let $B=\{2x,3x+x^2,-1\},B^{\prime }=\{1,1+x,1+x+x^2\}$
Need to find the transformation matrix from $B$ to $B^{\prime }$.
I know that:
$(a{x}^2+bx+c{)}_B=(\frac{b-3c}{2},c,-a)$
$(a{x}^2+bx+c{)}_{B^{\prime }}=(a-b,b-c,c)$
How to proceed using this info in order to find the transformation matrix?

Is a matrix multiplied with its transpose something special?
In my math lectures, we talked about the Gram-Determinant where a matrix times its transpose are multiplied together.
Is ${\displaystyle {\mathrm{\forall }}^T}$ something special for any matrix A?