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I will surmise what I've learned from a couple of Wikipedia articles below:

Vector projection:

The vector projection of a vector $\vec{a}$ onto a vector $\vec{b}$ (also known as the vector component or vector resolution of $\vec{a}$ in the direction of $\vec{b}$) is defined as the following:

$$\text{proj}_\vec{b}\vec{a}=\frac{\vec{a}\cdot \vec{b}}{|\vec{b}|}\hat b$$

, where $|\vec{b}|$ is the length of $\vec{b}$, $\hat{b}$ is the unit vector in the direction of $\vec{b}$, and the operator $\cdot$ denotes a dot product.

Scalar projection:

The scalar projection (also known as scalar component) of a vector $\vec{a}$ onto a vector $\vec{b}$ is given by the following:

$$s=|\vec{a}|\cos\theta$$

, where $|\vec{a}|$ is the length of $\vec{a}$, and $\theta$ is the angle between $\vec{a}$ and $\vec{b}$.

My question:

  1. Evidently, a vector projection/component is a vector and a scalar projection/component is a scalar. In this top answer, @RonGordon promulgates the component of $\vec{a}$ along $\vec{b}$ as a scalar. Is it a convention to assume that "component" means scalar component/projection, just as @RonGordon did, unless it is explicitly specified otherwise (by the writing of vector projection)? In other words, if I find "vector projection" written anywhere, it will be sufficiently clear what the author means. Similarly, if I find "scalar projection" written, it will be sufficiently clear what the author means as well. However, if I find only "component" written somewhere, what should I interpret it as?
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    $\begingroup$ Yes and also, the Wiki you linked confirms this too. Go to the Scalar Projection link and look at the 5th or 6th line starting with "multiplying the scalar projection by..." $\endgroup$ May 14 at 5:25
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    $\begingroup$ Yep, you're right. The scalar projection is the "scalar part" and you times it by the unit vector in the direction of $b$. I should have added notation to make that more clear! $\endgroup$ May 14 at 5:27
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    $\begingroup$ Also, if you're ever in doubt: remember the dot product always gives a scalar. So $\frac{a\cdot b}{|b|}$ is always a scalar. $\endgroup$ May 14 at 5:28
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    $\begingroup$ Ah yes, "scalar projection" is a slightly unfortunate and unclear phrase, "scalar component" is better in my opinion. $\endgroup$ May 14 at 5:31
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    $\begingroup$ I don't think a definitive answer to this question exists. I think what a "component" is will depend on context. In a change of basis, a component is a scalar. In a force decomposition diagram in a physics text you might see components that are vectors. Or the components might be described by magnitude, i.e., scalars. It will depend on how the "components" were found and what we intend to do with them. $\endgroup$
    – David K
    2 days ago

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The two values you call vector projection and scalar projection are more tightly linked than your deifinitions seem to imply. In fact, by definition, we have that for two vectors $\vec a,\vec b$, the cosine of the angle between the two vectors is defined to be $$\cos\theta = \frac{\vec a\cdot \vec b}{|\vec a||\vec b|}$$

which means that what you call "the scalar projection of $\vec a$ along $\vec b$", which is $|\vec a|\cos\theta$, is in fact equal to $$|\vec a|\cos\theta = |\vec a| \frac{\vec a\cdot \vec b}{|\vec a||\vec b|} = \frac{\vec a\cdot \vec b}{|\vec b|}.$$

Note that this value is precisely the length of what you call the "vector projection of $\vec a$ along $\vec b$". Also, since the length of the vector is a scalar, while the vector projection itself is a vector, it is very common to just use the word "projection" in common mathematical language, because in the vast majority of cases, it is clear from context whether we are talking about a vector (and thus a vector projection) or a scalar.


Additionally, there is the word "component". This word will most almost always denote a scalar quantity, and is tightly linked to the concept of projections. In particular, in $\mathbb R^n$, the $i$-th component of a vector $x\in\mathbb R^n$ is equal to the scalar projection of $x$ along the $i$-th basis vector.

This idea can be greatly generalized. If you are familiar with what a basis of a vector space is, then you might remember that if $B=\{b_1,\dots,b_n\}$ is a basis for a vector space $X$, then any $x\in X$ can be uniquely represented as $x=\alpha_1b_1+\cdots+\alpha_n b_n$. The common terminology is to refer to the values $\alpha_1,\dots,\alpha_n$ (which are scalars) as the components of $x$ in the basis $B$.

Now, you might already be sensing the connection between scalars projections and components. Indeed, if $B$ is an orthogonal set, then $\alpha_i$ is precisely equal to the scalar projection of $x$ along the vector $b_i$.


So, with all that out of the way, you can look again at the post you link as your question. The accepted answer states that "$\dfrac{\vec a \cdot \vec b}{|b|}$ is the component of $\vec a$ along $\vec b$." What this sentence is saying is basically "In any orthogonal basis in which $b$ is one of the basis vectors, $\dfrac{\vec a \cdot \vec b}{|b|}$ is the component of $a$ belonging to $b$ in that basis".

In that sense, the answer you link is correct, as long as you understand the implicit facts behind it :).

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The scalar projection of $\vec{a}$ on $\vec{b}$ is just the magnitude of the (vector) projection of $\vec{a}$ on $\vec{b}.$

Referring to your fourth comment above, “vector component” is ambiguous to my ear: you use it there to mean some component that is a vector, but it also sounds like it means some scalar component of a vector. As for “scalar component”, I prefer just saying “scalar projection” or “projection length” or “length of projection”.

Analogously, the scalar rejection of $\vec{a}$ from $\vec{b}$ is just the magnitude of the (vector) rejection of $\vec{a}$ from $\vec{b}.$

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