Find the dimension of each of the following vector spaces.

An object moving in the xy-plane is acted on by a conservative force described by the potential energy function ${\displaystyle U(x,y)=\alpha (\frac{1}{x^2}+\frac{1}{y^2})}$ where ${\displaystyle \alpha }$ is a positive constant. Derivative an expression for the force expressed terms of the unit vectors ${\displaystyle \overrightarrow{i}}$ and ${\displaystyle \overrightarrow{j}}$.

I need to find a unique description of Nul A, namely by listing the vectors that measure the null space
$A=\left[\begin{array}{ccccc}1& 5& -4& -3& 1\\ 0& 1& -2& 1& 0\\ 0& 0& 0& 0& 0\end{array}\right]$

T must be a linear transformation, we assume. Can u find the T standard matrix.${\displaystyle T:{\mathbb{R}}^2\to {\mathbb{R}}^4,T\left(e_1\right)=(3,1,3,1)\text{and}T\left(e_2\right)=(-5,2,0,0),where{e}_1=(1,0)\text{and}{e}_2=(0,1)}$

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.

Consider two vectors A=3i - 1j and B = - i - 5j, how do you calculate A - B?

Let vectors A=(1,0,-3) ,B=(-2,5,1) and C=(3,1,1), how do you calculate 2A-3(B-C)?

What is the projection of ${\displaystyle <6,5,3>}$ onto ${\displaystyle <2,-1,8>}$?

What is the dot product of ${\displaystyle <1,-4,5>}$ and ${\displaystyle <-5,7,3>}$?

Which of the following is not a vector quantity?
A)Weight;
B)Nuclear spin;
C)Momentum;
D)Potential energy

How to find all unit vectors normal to the plane which contains the points ${\displaystyle (0,1,1),(1,-1,0)}$, and ${\displaystyle (1,0,2)}$?

What is a rank $1$ matrix?

How to find unit vector perpendicular to plane: 6x-2y+3z+8=0?

Can we say that a zero matrix is invertible?

How do I find the sum of three vectors?

How do I find the vertical component of a vector?