I'm not sure if this question is asking me to find one vector that fulfils both of these criteria or one vector for each criteria. I have tried solving for one vector that satisfies both criteria but it doesn't seem possible. I will try to explain why below.
Identify the vector components of a f parallel and orthogonal to d.
f = (2,1,1) d = (1,2,3)
I know that in order for a vector to be parallel to f, it has to be a scalar multiple of f.
V = (2,1,1)λ
where λ is any non-zero scalar.
I know that in order for a vector to be perpendicular to d, the dot product of the vector and d must evaluate to zero.
Lets say V = (a,b,c)
(a,b,c)·(1,2,3) = 0
a + 2b + 3c = 0
Right so this is where I feel as if it is impossible to have a single vector that fulfils both criteria.
The components of f are all positive. So if i try to evaluate a + 2b + 3c = 0, to find a vector that is a scalar multiple of f, each components of this vector cannot be all negative or all positive. Otherwise it will never evaluate to zero. But in order for it to be parallel to f, each component must be all negative or all positive. I am extremely confused, please help me understand this confusing question.
As noticed in the comments, it seems we need to find the components of $f$ parallel and orthogonal to $d$ that is
$f_{\parallel} = \frac{f\cdot d}{|d|^2}d$
$f_{\perp} =f-f_{\parallel}=f- \frac{f\cdot d}{|d|^2}d$