Is the component of a vector along another vector also a vector?

Question

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?

Answer 1

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.

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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 $b_i$.

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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 :).

Answer 2

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 $\vec{b}.$