Currently learning about direction cosines and I'm feeling a bit confused. Hopefully you can help.
Let's say we have a vector V in 3-dimensions. V makes an angle of a with the X-axis, b with the Y-axis, and c with the Z-axis.
Shouldn't a + b = 90°? When I think of it from a 'bird's eye' perspective, it seems like it should be. My book is saying that isn't true though.
The problem here is there are two ways to interpret the statement "$V$ makes an angle of $a$ with the $X$ axis". The way it seems you're interpreting it is the following:
In which case I agree that $a+b=90$ (when $a$ and $b$ are defined and are acute; if $V$ is along the $z$-axis then neither angle is defined since the projection to the $xy$-plane sends $V$ to $0$). When viewing things from a bird's-eye perspective what you're doing is projecting onto the $xy$-plane, so this is how you're defining the angle between $V$ and the $x$ axis. However what the angle between $V$ and the $x$ axis actually means is the following:
That is, you take the angle between $V$ and the $x$ axis in the plane containing both $V$ and the $x$ axis. In general, taking the angle between two vectors means taking the angle between them in the plane which contains both of them.