Map vector A onto line l would that mean the projection of A

In summary, the conversation discusses the difference between mapping a vector onto a line and projecting or rotating it onto the line. It is stated that "map vector v onto line l" does not necessarily mean either of those, but rather refers to a function that changes vector v to part of line l. It is suggested that using the phrases "project vector v onto line l" or "rotate vector v onto line l" would be more accurate. The conversation then goes on to discuss a simple solution for obtaining a standard matrix for the linear transformation of any vector onto the line. This solution involves using a matrix with all elements equal to 1, which maps the vector onto the line but is not a projection or rotation.
  • #1
Oerg
352
0
If i were to say: Map vector A onto line l

would that mean the projection of A onto l or the rotation of A onto l?
 
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  • #2


"Map vector v onto line l" doesn't necessarily mean either of those- it just means that some function changes vector v to part of line l.

I would say either "project vector v onto line l" or "rotate vector v onto line l".
 
  • #3


Then what if we want a standard matrix for the linear transformation for any vector onto the line? let's say we have

[tex]x=t[/tex]
[tex]y=t[/tex]
[tex]z=t[/tex]

then a simple solution would be

[tex][T]=\left[ \begin{array}{ccc}
1 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 1
\end{array} \right][/tex]

because

[tex]T(e_1)=\left[\begin{array}{ccc}
1\\1\\1 \end{array}\right][/tex]

[tex]T(e_2)=\left[\begin{array}{ccc}
1\\1\\1 \end{array}\right][/tex]

[tex]T(e_3)=\left[\begin{array}{ccc}
1\\1\\1 \end{array}\right][/tex]

this seems like a pretty simple solution, it maps the vector onto the line, but it is not a projection or a rotation.
 

1. What is the meaning of "mapping vector A onto line l"?

Mapping vector A onto line l means finding the projection of vector A onto line l. This involves finding a vector that is parallel to line l and has the same direction as vector A.

2. How is the projection of vector A onto line l calculated?

The projection of vector A onto line l can be calculated using the dot product formula: projlA = (A · ⅼ) * ⅼ, where ⅼ is a unit vector parallel to line l.

3. What does the projection of vector A onto line l represent?

The projection of vector A onto line l represents the component of vector A that lies along line l. It can also be thought of as the shadow of vector A cast onto line l.

4. Can the projection of vector A onto line l be negative?

Yes, the projection of vector A onto line l can be negative if vector A and line l are pointing in opposite directions.

5. How is the projection of vector A onto line l useful?

The projection of vector A onto line l can be useful in various applications, such as computer graphics and physics, where it is used to determine the magnitude of a vector in a specific direction. It can also be used to solve problems involving vectors and lines, such as finding the shortest distance between a point and a line.

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