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1.1: Scalars vs Vectors

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  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    We are all familiar with the notion of speed, i.e. how fast something is going. Velocity is the exact same thing, except we have specified a direction. In technical terms, speed is a scalar, whereas velocity is a vector. We shall further explore the idea of a vector in the coming pages. For now, imagine it to be arrow of a given length that is pointing in a certain direction.

    Given this information, you might wonder: how do we take into account the direction when solving problems? This is where the elegance of the vector is apparent. We can arbitrarily decide a "forward " direction and a "backward" direction on an axis, i.e. a straight line. Conventionally, forward is denoted by a positive value and backward by a negative value, but this too is arbitrary. All you have to do is ensure you are following the same convention throughout.


    1.1: Scalars vs Vectors is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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