Let W be the set of all vectors of the form shown, where a, b, and c r

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

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 ${\displaystyle |u\cdot v|}$ and determine whether ${\displaystyle u\cdot v}$
is directed into the screen or out of the screen.
${\displaystyle |u\cdot v|=??}$

Label the following statements as being true or false.
(a) There exists a linear operator T with no T-invariant subspace.
(b) If T is a linear operator on a finite-dimensional vector space V, and W is a T-invariant subspace of V, then the characteristic polynomial of Tw divides the characteristic polynomial of T.
(c) Let T be a linear operator on a finite-dimensional vector space V, and let x and y be elements of V. If W is the T-cyclic subspace generated by x, W

(a) find the projection of u onto v and (b) find the vector component of u orthogonal to v. u = ⟨6, 7⟩, v = ⟨1,4⟩

Is the vector space C∞[a,b] of infinitely differentiable functions on the interval [a,b], consider the derivate transformation D and the definite integral transformation I defined by D(f)(x)=f′(x)D(f)(x)=f′(x) and I(f)(x)=∫xaf(t)dt f(t)dt. (a) Compute (DI)(f)=D(I(f))(DI)(f)=D(I(f)). (b) Compute (ID)(f)=I(D(f))(ID)(f)=I(D(f)). (c) Do this transformations commute? That is to say, is it true that (DI)(f)=(ID)(f)(DI)(f)=(ID)(f) for all vectors f in the space?