Finding the angle a vector makes with the 'horizontal'
By John Thompson
Suppose a cable car moves with direction vector $\vec{d}$ = $$\begin{pmatrix} 2\\ -2\\ 1\\ \end{pmatrix} $$The question asks, "at what angle to the horizontal does the cable car travel?". By horizontal, if you assume the x-axis, the angle can be found using $$\frac{|\vec{a}\cdot\vec{b}|}{|\vec{a}\vec{b}|}$$) Using this, the angle I find between $\vec{d}$ and $$\begin{pmatrix} 2\\ 0\\ 0\\ \end{pmatrix} $$The angle I find is $48.2^\circ$. If I use the z-vector instead:$$\begin{pmatrix} 0\\ 0\\ 1\\ \end{pmatrix}$$The angle I find is $70.5^\circ$
The answer is $\approx 19.5^\circ$. How would I get this value?
Additional information: The cable car starts at $(10,3,0)$ and it moves in the direction of $\vec{d}$ at a speed of $4.5 \;m\,s^{-1}$
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$\begingroup$You have to find the angle between $\vec{d}$ and the projection of $\vec{d}$ onto the horizontal plane, which is the vector $\vec{p}=(2,-2,0)$. Try to apply the same equation with those two vectors.
$\endgroup$ 2 $\begingroup$'Horizontal' here means the $x,y$ plane.
If you sketch a picture, it could be clear that the angle of a vector and a plane and the angle with a normal vector of the plane adds up to $90^\circ$.
So, since you already found the angle with the $z$-axis to be $70.5^\circ$, the angle to the horizontal will be $90^\circ-70.5^\circ=19.5^\circ$.
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