Does this question make sense?

Question

Problem :

a. Using the inner product, develop formulas to find the components of a vector $f$ perpendicular and parallel to a vector $d$.

b. Consider a force $f=\left[\begin{array}{l}3 \\ 1 \\ 2\end{array}\right].$

Use the results of part (a) to find the components of $f$ perpendicular and parallel to the vector $d=\left[\begin{array}{l}1 \\ 1 \\ 0\end{array}\right]$

I have two problems with this problem :

1. How can a vector be perpendicular and parallel to another vector at the same time (unless it's a zero vector) ?
2. In the example in the problem, the two vectors are literally neither perpendicular nor parallel, what is really meant by the question ?

This was a HW given to a friend, I keep telling them there must be mistakes but they insist that there is none.

Answer 1

The task is to decompose a vector $f$ into two components: one that is perpendicular to $d$ and one that is parallel to $d$. That is, you are looking for two vectors $f_1$ and $f_2$ such that

1. $f = f_1 + f_2$,
2. $f_1$ is perpendicular to $d$,
3. $f_2$ is parallel to $d$.