In geometric terms ...

In a parallel projection, points are projected (onto some plane) in a direction that is parallel to some fixed given vector.

In an orthogonal projection, points are projected (onto some plane) in a direction that is normal to the plane.

So, all orthogonal projections are parallel projections, but not vice versa. A parallel projection that is not an orthogonal projection is called an "oblique" projection.

This could all be translated into the language of linear algebra, I suppose, but I don't think that would make it any clearer.

Let's restrict our attention to subspaces $V$ of $\mathbb{R}^3$ rather than $\mathbb{R}^n$. Once this case is understood, you can try to generalize it. It is important to think slowly from the definitions. Geometric intuition will come afterwards (and be correct). I will not recall the definition of orthogonal projection onto a subspace for you, you can look that up in your notes/textbook.

You appear to be confusing several concepts. Let me try to clarify them for you.

Fix a subspace $V \subseteq \mathbb{R}^3$ (this could be the origin, a line through the origin, a plane containing the origin, or the entire space $\mathbb{R}^3$). Let $T_V\colon \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be *the linear transformation defined by orthogonal projection onto the subspace* $V$.

Any linear transformation has a kernel and an image. They are defined for $T_V$ as follows:

$$\text{image}(T_V) = \left\{ y \in \mathbb{R}^3 \colon \exists x \in \mathbb{R}^3 \text{ such that } T_V(x) = y \right\} $$

$$\text{kernel}(T_V) = \left\{ x \in \mathbb{R}^3 \colon T_V(x) = 0 \right\}$$

(you may note that both the image and the kernel of $T_V$ are subspaces of $\mathbb{R}^3$).

From the first definition, we can explain that $$\text{image}(T_V) = V.$$ The proof uses two key facts: the definition of the image of a linear transformation, and the definition of the map $T_V$.

*Proof that $\text{image}(T_V) = V$:* In order to do this, we show that $\text{image}(T_V) \subseteq V$ and $V \subseteq \text{image}(T_V)$:

For any vector $x \in \mathbb{R}^3$, the orthogonal projection of $x$ onto $V$ is an element of $V$. Thus $\text{image}(T_V) \subseteq V$.

On the other hand, if $x$ is an element of $V$, then $T_V(x) = x$ (the orthogonal projection of a vector in $V$ onto $V$ is itself), so $V\subseteq \text{image}(T_V)$. This completes the proof. $\square$

Thus:

- If $V$ is a line in $\mathbb{R}^3$, then $\text{image}(T_V)$ is the same line.
- If $V$ is a plane in $\mathbb{R}^3$, then $\text{image}(T_V)$ is the same plane.
- If $V$ is the entire space $\mathbb{R}^3$, then $\text{image}(T_V)$ is the entire space $\mathbb{R}^3$.

Now we would like to describe the second space, $\text{kernel}(T_V)$. In order to do this, it is useful to recall that the *orthogonal complement of a subspace* $V$ is a new subspace defined in the following way:
$$V^{\perp} = \left\{ y\in \mathbb{R}^3 : \forall x\in V, \langle x,y\rangle = 0 \right\}.$$
In plain English, $V^{\perp}$ is the set of all vectors that are orthogonal to every vector in $V$.

You should think about why the following statements are true (note that tehy only make sense if $V$ is a subspace of $\mathbb{R}^3$):

- If $V$ is a line, then $V^{\perp}$ is a plane.
- If $V$ is a plane, then $V^{\perp}$ is a line.
- If $V$ is the origin, then $V^{\perp}$ is the entire space $\mathbb{R}^3$.
- If $V$ is the entire space $\mathbb{R}^3$, then $V^{\perp}$ is the origin.

You should also try to draw pictures of some examples.

The following statement contains the intuition you are after. I will leave the proof of this to you.

$$V^{\perp} = \text{kernel}(T_V).$$

## Best Answer

The general case of projections is this:

Suppose a vector space $V$ decomposes as a direct sum of subspaces $U$ and $W$, i.e., $V = U \oplus W$. Then every $v \in V$ can be expressed uniquely as $v = u + w$, where $u \in U$ and $w \in W$. We say that the vector $u$ is the

projectionof the vector $v$ onto the subspace $U$ (with respect to the decomposition $U \oplus W$).Notice that this definition has nothing to do with orthogonality! We can project onto any subspace with respect to any decomposition.

Now let's look at orthogonal projections. Assume, as above, $V = U \oplus W$, but now additionally we assume that $U$ and $W$ are orthogonal. (Here I must assume that $V$ is now an inner product space, so we have a notion of orthogonality. If you are not familiar with such spaces, you can just think of $\mathbb{R}^n$ with the usual notion of orthogonality.)

We designate one of the subspaces (say $U$) as the "parallel" one we are projecting onto, and the other one ($W$) as the "orthogonal" one. (I am using the terminology of the question you linked to.) Then, in your examples 1-3 above: