Can we find an example where $\mathbb{R}^3$ is a direct sum of two subspaces that are not orthogonal?
A vector space $X$ is said to be a direct sum of two of subspaces $Y$ and $Z$ of $X$ if each $x \in X$ can be expressed uniquely as a sum $$x = y + z,$$ where $y \in Y$ and $z \in Z$.
And, if, in addition, $X$ is also an inner product space, then the subspaces $Y$ and $Z$ are said to be orthogonal if $$\langle y, z \rangle = 0$$ for all $y \in Y$ and for all $z \in Z$.
Is there any general result of this sort about finite-dimensional inner product spaces?
Take plane and a straight lines in $\mathbb{R^3}$ passing through origin such that the line is not perpendicular to the plane and obviously not in the plane.
There is no need to do such a calculative thing. An easy way out is the following:
Just take the $xy$ plane in $\mathbb{R^3}$ (you know its equation) and write down the equation of the straight line passing through $(1,0,1)$ and origin and lying in the $xz$ plane ($t\to (t,0,t)$)