The orthogonal projection of u onto v is 0? u = $\begin{bmatrix}-1\\1\\0\\1\\0\end{bmatrix}$ and v = $\begin{bmatrix}1\\0\\1\\1\\1\end{bmatrix}$
Find proj_v^u.
The orthogonal projection of u onto v is equal to (u ∙ v/ v∙v)(v)
u ∙ v = 0. Can the orthogonal projection be equal to zero? How can I visualize this?
 A: Yes, the projection of $u$ onto $v$ can be $0$. The projection of $u$ onto $v$ is the vector of the form $\lambda v$ with smaller distance to $u$. So, asserting that the projection of $u$ onto $v$ is $0$ simply means that of all vectors of the form $\lambda v$, the one which is closest to $u$ is the one for which $\lambda=0$. Geometrically, this means that $u$ and $v$ are orthogonal.
A: Here is a good visualization of the dot product as a projection can be found here.  The whole "essence of linear algebra" series is actually fantastic.
When you watch the video you can see that the dot product of a vector $\vec v$ with  a unit vector $\hat u$ can be seen as the length of the projection of $\vec v$ onto $\hat u$.  If the dot product is positive, it means the projection of $\vec v$ lands in the same direction as $\hat u$.  If the dot product is negative, it means the projection of $\vec v$ lands in the opposite direction as $\hat u$.  If the dot product is zero, it means the projection became the zero vector, which means $\vec v$ had to be perpendicular to $\hat u$.
A: It can be zero, and in fact all linear projections will have an entire subspace of vectors which are all equal to zero under the projection. You can see this in your example by taking a scalar multiple of $u$ and noting the dot project remains zero. This subspace will always be orthogonal to the space you're projecting onto, in this case the space spanned by $v$. Visualizing orthogonal subspaces isn't too hard in $\mathbb{R}^3$. You have a plane through the origin which is perpendicular to a line going through the origin. You can either project the plane onto the line, or the line onto the plane with the entire line or plane being mapped to zero under the projection. This is the main model I use for visualizing quotient spaces as well.
