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I know that components of vectors are defined as being scalars. For example

        ax= a*cos(x)

Also I know that scalars cannot have a directions. That is OK, but what is happening when we take 180°>x>90° like in photo. What should I understand when I see a negative scalar. It feels like it has direction.How it doesn't?

Alim Karaçay
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3 Answers3

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A scalar is just a regular number, and thus it can be positive or negative. Scalars themselves don't have directions but they define the direction of the vector that they generate.

agaminon
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$\vec a_{\rm x} = -3\hat i$ can be interpreted in two ways.

$\vec a_{\rm x} = -3\,(+\hat i)$
Here the vector has a magnitude of $3$ - magnitudes are always positive, think of it as the length of the vector when drawn on a piece of paper.
The vector has a component of $-3$ in the $\hat i$ direction - in this case you have chosen a particular direction which is the direction of increasing $x$, $\hat i$, and the vector $\vec a_{\rm x}$ points in the opposite direction to $\hat i$.

$\vec a_{\rm x} = 3\,(-\hat i)=3\,( \hat {i'})$
As before here the vector has a magnitude of $3$.
The vector has a component of $+3$ in the $(-\hat i)$ or $(\hat {i'})$ direction - in this case you have chosen a particular direction which is the direction of decreasing $x$, $-\hat i$ or $(\hat {i'})$, and the vector $\vec a_{\rm x}$ points in the same direction as $(-\hat i)$ or $(\hat {i'})$.

Farcher
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A scalar can totally be signed, and its minus sign indicates direction, which may or may not be spatial.

For example, the sentence "the speed of that object, which is at $-3$ coulombs and $-4^\circ C,$ is changing at $-5$ m/s$^2$" contains three scalars, only the third of which has a spatial direction.

ryang
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