Help and Resources to help with vector application problems I'm currently studying vectors through an online university and while I have a good grasp of the basics of vectors, I am completely dumbfounded by its applications. Here is the problem in my book:
The thrust of an airplane's engine produces a speed of 600 mph in still air. The plane is aimed in the direction of (2,2,1) and the wind velocity is (10,-20,0) mph. Find the velocity vector of the plane with respect to the ground and find the speed. 
1) I'm firstly confused by the wording: how do you have a velocity vector "with respect to the ground"? Is it asking me to use the ground as a frame of reference, ie, how is altitude changing? I'm completely confused by this. 
2) Do you recommend any resources that I can use? I know the basics like how to find magnitude, dot/cross product, etc. I just don't know how to use them when I have to. 
Thanks. 
 A: To be perfectly honest, I'm not exactly sure what to make of the phrase "with respect to the ground," but I don't think it's crucial to solving the problem. I'm sure someone here could give a better explanation of what exactly it means.

As for the problem, you'll need to add the plane's velocity vector to the wind's velocity vector to determine the plane's net velocity vector. Since you don't have the components of the plane's velocity vector, you'll need to compute them. Fortunately, the problem tells you both the direction and magnitude of the plane's velocity vector, so it won't be too hard to calculate.
The problem tells us that the plane's velocity vector, $\mathbf{v}$, is in the direction of the vector $<2,2,1>$ and has a magnitude of $600$. To find the component form of this vector, we need a unit vector in the direction of $\mathbf{v}$ whose components we can then multiple by $600$ to find $\mathbf{v}$ itself. Finding a unit vector is just a matter of dividing a vector by its length, so if we call $\hat{\mathbf{v}}$ the unit vector in the direction of $\mathbf{v}$, then $$\hat{\mathbf{v}}=\frac{1}{\sqrt{2^2+2^2+1^2}}\begin{bmatrix}2\\2\\1\end{bmatrix}=\frac{1}{\sqrt{4+4+1}}\begin{bmatrix}2\\2\\1\end{bmatrix}=\frac{1}{3}\begin{bmatrix}2\\2\\1\end{bmatrix}$$
As we already know, $\mathbf{v}$ is a vector in the same direction as $\hat{\mathbf{v}}$ that has length $600$, so $$\mathbf{v}=600\hat{\mathbf{v}}=\frac{600}{3}\begin{bmatrix}2\\2\\1\end{bmatrix}=200\begin{bmatrix}2\\2\\1\end{bmatrix}=\begin{bmatrix}400\\400\\200\end{bmatrix}$$
Now that we know $\mathbf{v}$, we can add it to the velocity vector of the wind, $\mathbf{w}$, to find the plane's net velocity vector, which we can call $\mathbf{u}$. $$\mathbf{u}=\mathbf{v}+\mathbf{w}=\begin{bmatrix}400\\400\\200\end{bmatrix}+\begin{bmatrix}10\\-20\\0\end{bmatrix}=\begin{bmatrix}410\\380\\200\end{bmatrix}$$

As for resources for applications of vectors, there are a number of ways you could go. I've always found Khan Academy to be enormously helpful, and they've got wonderful videos on physics and linear algebra, both of which are fields that abound with vectors. If you're more interested in books, John R. Taylor's Classical Mechanics is a thorough introduction to mechanics, full of all different kinds of vector operations. I would also highly recommend Jeffrey Holt's Linear Algebra with Applications, since that was my first real introduction to vectors, and I was quite happy with it.
