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 $<4,-\frac{16}{3},-2>$

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

Label the following statements as being true or false.
(a) If V is a vector space and W is a subset of V that is a vector space, then W is a subspace of V.
(b) The empty set is a subspace of every vector space.
(c) If V is a vector space other than the zero vector space {0}, then V contains a subspace W such that W is not equal to V.
(d) The intersection of any two subsets of V is a subspace of V.
(e) An ${\displaystyle n\times n}$ diagonal matrix can never have more than n nonzero entries.
(f) The trace of a square matrix is the product of its entries on the diagonal.

Without calculation, find one eigenvalue and two linearly independent eigenvectors of
$A=\left[\begin{array}{ccc}5& 5& 5\\ 5& 5& 5\\ 5& 5& 5\end{array}\right]$
Justify your answer.

In figure B, a standard man (mass = 70 kg) uses crutched. The crutches each make an angle of  = 250 with the vertical. Half the standard man’s weight is supported by the crutches; the other half is supported by the normal force acting on the sole of the feet. Assuming the standard man is motionless, calculate the magnitude of the force supported by each crutch

I'm having trouble with these types of questions. I have the following vector ${\displaystyle u=(4,7,-9)}$ and it wants me to find 2 vectors that are perpendicular to this one.
I know that ${\displaystyle (4,7,-9)\cdot (x,y,z)=0}$

Consider ${\displaystyle V=\cos \left(x\right),\sin \left(x\right)}$ a subspace of the vector space of continuous functions and a linear transformation ${\displaystyle T:V\to V}$ where ${\displaystyle T\left(f\right)=f\left(0\right)\times \cos \left(x\right)-f\left(\pi 2\right)\times \sin \left(x\right).}$

Find the matrix of T with respect to the basis ${\displaystyle \cos \left(x\right)+\sin \left(x\right),\cos \left(x\right)-\sin \left(x\right)}$ and determine if T is an isomorphism.