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I've just started studying linear algebra, and I'm having trouble understanding something that apparentely seems obvious to everyone else. My question is, why does multiplying a vector by a positive scalar does not change its direction? I've seen some visual representations of this and it does seem reasonable to me that that should be the case. However, I wonder whether there is a way to actually show it mathematically? In the 2d coordinate plane, for instance, I guess way one of showing that vector $A$ has the same direction of $cA,c>0$ would be to show that the line containing $A$ has the same slope as the one containing $cA$ (I'm not sure if this approach is correct though). However, how to proceed in the case of $n$ dimensions? I think I must be missing out something basic here, but I really can't seem to get my head around it. Any help will be highly appreciated. Thanks very much in advance.

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    $\begingroup$ First ask yourself what "direction" even means in $n$ dimensions. One definition is that... two vectors point in the same direction if one is a positive scalar multiple of the other! $\endgroup$ – Qiaochu Yuan Dec 5 '17 at 1:18
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It is because of similar triangles.

Consider all possible triangles that the vector makes with the coordinate axes. When all the lengths are multiplied by $c$, the angles of the transformed triangles remain the same because the lengths are proportional.

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For each $\alpha > 0$ $$\text{angle}(\mathbf{x}, \, \alpha \mathbf{x}) = \arccos \frac{(\mathbf{x}, \, \alpha \mathbf{x})}{\|\mathbf{x}\| \, \|\alpha \mathbf{x}\|} = \arccos\frac{\alpha (x,x)}{\alpha \|\mathbf{x}\| \, \| \mathbf{x} \|} = \text{angle}(\mathbf{x}, \, \mathbf{x}) = 0$$

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One way you might think of defining "direction" in a space of general dimension $n\geq 1$ is the set of vectors $u$ with $\|u\|_{2}=1.$ In two dimensions, this is just the unit circle. If we accept this definition, then we can write any nonzero vector $v\in\mathbb{R}^{n}$ as $$v=\|v\|_{2}\left(\frac{v}{\|v\|_{2}}\right),$$ where the vector in parentheses is the "direction" of $v$ in the sense defined above, and $\|v\|_{2}$ is a positive scalar. But this makes it clear that if $a>0,$ then $$av=(a\|v\|_{2})\left(\frac{v}{\|v\|_{2}}\right)$$ has the same direction vector, but a different positive scalar multiplying it. In this sense, they have the same direction.

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To what degree do the truth of the following claims seem clear to you?

  1. Adding a vector to itself results in a vector pointing in the same direction.
  2. Multiplying a vector by a positive integer results in a vector pointing in the same direction.
  3. Multiplying a vector by a positive rational number results in a vector pointing in the same direction.
  4. Multiplying a vector by a positive real number results in a vector pointing in the same direction.
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