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I'm relatively new to linear algebra. I've got a question about a question. I'm not looking for the exact answer (I'll try to find it by myself). I stumbled upon the following question :

"*At 12:00 pm, a spaceship is at position $$\begin{pmatrix} 3 \\ 2\\ 4 \end{pmatrix}$$km away from the origin with respect to some 3 dimensional co ordinate system. The ship is travelling with velocity $$\begin{pmatrix} -1 \\ 2\\ -3 \end{pmatrix}$$km/h What is the location of the spaceship after 2 hours have passed?".

I understand what a $$\begin{pmatrix} 3 \\ 2\\ 4 \end{pmatrix}$$ position means (in three dimensional Cartesian coordinate system with x, y, z axes). I can point precisely the position on a paper. But what does a " [-1,2,-3] velocity " mean ?

Again, I'm not looking for the precise answer. I'd just like to know how to approach the problem since a same format (vectors) is used to tackle two different concepts (namely, position and velocity). How can I process to some calculations in this "position" I would say, aha.

Thank you :)

Alex

aschiff
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    In this setting, "$[-1, 2, -3]$ km/h velocity" means it is traveling $\sqrt{(-1)^2 + 2^2 + (-3)^2}$ km/h in the direction that the vector $[-1, 2, -3]$ is pointing. – angryavian Dec 28 '19 at 21:03
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    It means that after k hrs its position is $(3, 2, 4)+k(-1, 2, -3)$. – almagest Dec 28 '19 at 21:09
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    $-1$ km/hr in the x direction, $2$ km/hr in the y direction and$ -3$ km/hr in the z direction. So after $2$ hours add each velocity component times $2$ to each of the corresponding coordinates in the position vector. – Phil H Dec 28 '19 at 21:12

2 Answers2

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A vector can represent a position by thinking of it as specifying the displacement from the origin of your system of coordinates to the position in question -- hence such vectors are called position vectors, not strangely.

Since velocity by definition is the derivative of position, then velocity as well is a vector, derived by differentiating the position vector.

Continuing this train of thought, you can see why acceleration, force (derivative of momentum, which involves velocity), etc. are all vectors.

Allawonder
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Answer spoiler!

We are already given a 3-dimensional position. From that position, we move to $\begin{pmatrix}-1 \\ 2 \\ -3\end{pmatrix}$ per hour. So, to figure out the position after two hours travelling this velocity, write $\begin{pmatrix}3 \\ 2 \\ 4\end{pmatrix} + 2 * \begin{pmatrix}-1 \\ 2 \\ -3\end{pmatrix}$.

Community
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Martin Pekár
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