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October 23rd, 2015, 09:05 AM   #1
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Difference between projection and component

Hello,

I found out that what I am calling "projection" is actually called a "component", and the difference between what I call "projection" to a real projection is that that a projection is a component multiplied by the unit vector in the direction.

I wanted to know, why do we add this multipication of unit vector?
I know that the regular formula of component of a vector:
[A(dot)B] / (magnitude of A)
is because of the triangle formed with the angle (theta) between the two vectors. so geometrically, I know what component is, but up until today I called it projection.
So what is the explanation of adding the multipication-by-unit-vector part? Is there a geometrical explanation? Or any other explanation?

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October 23rd, 2015, 10:02 AM   #2
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Quote:
Originally Posted by noobinmath View Post
Hello,

I found out that what I am calling "projection" is actually called a "component", and the difference between what I call "projection" to a real projection is that that a projection is a component multiplied by the unit vector in the direction.
First, this is not true! Or, at least, you are leaving a lot out.
The "x-component" of a vector is, by definition, the "projection" of the vector onto the x-axis. Similarly, for the "y-component" and "z-component" relative to the projection of the vector onto the y and z axes respectively.
But we can have projections onto a arbitrary line, not necessarily coordinate axes.

Quote:
I wanted to know, why do we add this multipication of unit vector?
I know that the regular formula of component of a vector:
[A(dot)B] / (magnitude of A)
This is the formula for the "projection of vector B on vector A, not for "component of a vector".

Quote:
is because of the triangle formed with the angle (theta) between the two vectors. so geometrically, I know what component is, but up until today I called it projection.
So what is the explanation of adding the multipication-by-unit-vector part? Is there a geometrical explanation? Or any other explanation?

Thanks
As I said above, that dot product is for the projection of one vector on another. If that "other" is a unit vector along one of the coordinate axes, then it is, by definition, the component of the vector in the direction of the axis.
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October 24th, 2015, 06:16 PM   #3
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Thank you!
So tell me if I got this right:
The added multipication is only if we do not project the vector on one of the coordinate axes, but if we project it on another vector.
And even if we do project it on the coordinate axes, this multipication does exist, however it's not necessary because it equals to 1.

Btw, you get the magnitude of the projection, right?
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