Talk:Vector projection

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Using lowercase letter for scalar projection

Using a lowercase letter for scalar projection is quite common. I added a new section Vector projection#Notation in which I explained this notation. See also this text copied from Euclidean vector:

From Euclidean vector
Another way to represent a vector in n-dimensions is to introduce the standard basis vectors. For instance, in three dimensions, there are three of them:
${\displaystyle {\mathbf {e} }_{1}=(1,0,0),\ {\mathbf {e} }_{2}=(0,1,0),\ {\mathbf {e} }_{3}=(0,0,1).}$

These have the intuitive interpretation as vectors of unit length pointing up the x, y, and z axis of a Cartesian coordinate system, respectively. In terms of these, any vector a in ${\displaystyle \mathbb {R} ^{3}}$ can be expressed in the form:

${\displaystyle \mathbf {a} =\mathbf {a} _{1}+\mathbf {a} _{2}+\mathbf {a} _{3}=a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3},}$

where a1, a2, a3 are called the vector components (or vector projections) of a on the basis vectors or, equivalently, on the corresponding Cartesian axes x, y, and z (see figure), while a1, a2, a3 are the respective scalar components (or scalar projections).

In introductory physics textbooks, the standard basis vectors are often instead denoted ${\displaystyle \mathbf {i} ,\mathbf {j} ,\mathbf {k} }$ (or ${\displaystyle \mathbf {\hat {x}} ,\mathbf {\hat {y}} ,\mathbf {\hat {z}} }$, in which the hat symbol ^ typically denotes unit vectors). In this case, the scalar and vector components are denoted respectively ax, ay, az, and ax, ay, az (note the difference in boldface). Thus,

${\displaystyle \mathbf {a} =\mathbf {a} _{\text{x}}+\mathbf {a} _{\text{y}}+\mathbf {a} _{\text{z}}=a_{\text{x}}{\mathbf {i} }+a_{\text{y}}{\mathbf {j} }+a_{\text{z}}{\mathbf {k} }.}$

Paolo.dL (talk) 16:09, 24 April 2013 (UTC)

My problem with this notation is the text sometimes refers to a1. Is that the vector or the scalar? For the sake of readers completely unfamiliar with the subject, we should really improve notation so we aren't calling two different things by nearly the same name, because that's confusing.. many readers might not realise that bolding is significant. There's no harm is using an unambiguous name (it's not like there are widely accepted standards here anyway, since there are so many different possible contexts). Mark M (talk) 17:38, 24 April 2013 (UTC)
I added a new section Vector projection#Notation in which I explained this notation. People commonly uses it. Why should the readers be treated as dumb people? It is useful for them to learn this notation.
Similarly, it is confusing to use c or s for the scalar component of a. How do you call then the scalar component of a vector c or s? Every vector has its own components:
${\displaystyle \mathbf {a} =a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3},}$
${\displaystyle \mathbf {c} =c_{1}{\mathbf {e} }_{1}+c_{2}{\mathbf {e} }_{2}+c_{3}{\mathbf {e} }_{3},}$
${\displaystyle \mathbf {s} =s_{1}{\mathbf {e} }_{1}+s_{2}{\mathbf {e} }_{2}+s_{3}{\mathbf {e} }_{3}.}$
By the way, I disagree that the text uses a1 without calling it explicitly scalar projection and/or offering an evident comparison with the symbol used for vector projection.
In short, I don't think that this is confusing, I think it is useful and important to know.
Paolo.dL (talk) 10:30, 26 April 2013 (UTC)

Proposed merge

Essentially all of the content in Scalar projection is already in this article, so there's no need to have duplicate information. So a merge makes sense to me. Mark M (talk) 17:38, 24 April 2013 (UTC)

I agree, good idea. Enjoy yourself! M∧Ŝc2ħεИτlk 18:10, 24 April 2013 (UTC)
Scalar projection of a on b.
${\displaystyle s=|\mathbf {a} |\cos \theta =\mathbf {a} \cdot \mathbf {\hat {b}} ,}$