Question: > 
> At $t=0$, a point starts at $(10,10)$ with a velocity of $20$ units per second in the positive $x$ direction. It also has a constant acceleration of $5$ units per second squared towards to origin at all times. What is its parametric equation?
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First let me say that I am very unexperienced when it comes to differential equations, and I have no idea how to even approach this one.

To solve this problem, I started by saying that, if both $x$ and $y$ are functions of $t$,
$$x''=5\cos\arctan\frac{y}{x}=\frac{5x}{\sqrt{x^2+y^2}}$$
$$y''=5\sin\arctan\frac{y}{x}=\frac{5y}{\sqrt{x^2+y^2}}$$

Furthermore, because of the starting velocity, when $x''$ is integrated, the constant will end up being $20$. And when $x'$ and $y'$ are integrated, due to the starting position of the point, their constants will both be $10$. However, I don't know how to get to that point... I only know how to integrate *single* differential equations, and only simple ones at that. Can anybody explain to me if and how one can go about solving this problem?

Be warned: I made up this problem and I have no idea if it has a neat solution. If it can't be done and I am wasting my time, just tell me and explain why.

Answer: 
You can't: both matrices $Q$ and $S$ (and their transposes) are invertible, so their product is also invertible and hence can't be the zero matrix.

About the second question: The null space of $Q$ is $\{0\}$, so the null space matrix would be $[0]$ which is not orthonormal.
You can't. Any orthogonal matrix is nonsingular, and cannot have a product of zero with any nonzero matrix.