Does Vector Projection Depend on the Magnitude of Both Vectors?

[Zorodius]

Meaning of "projection"

Suppose you have two vectors, U and V.

Is it correct that the "projection" of vector U onto vector V is equal to U cos x, where U is the magnitude of vector U, and x is the angle between the two vectors? Specifically, is it correct that the projection of one vector onto another vector does not depend on the magnitude of the vector you are projecting on to?

If yes, is it also correct to say that the dot product does not represent a projection, unless you are projecting onto the unit vector?

 

[chroot]

The projection of u onto v is

$$\begin{equation*} \begin{split} \textrm{proj}_{\mathbf v} \mathbf u &= \frac{ \left| \mathbf u \cdot \mathbf v \right| }{ \left| \mathbf v \right|^2} \, \mathbf v\\ &= \frac{ \left| \mathbf u \right| \left| \mathbf v \right| \cos \theta}{ \left| \mathbf v \right|^2} \, \mathbf v \end{split} \end{equation*}$$

The projection does not depend on the length of the vector projected onto.

The dot product never represents a projection, because the dot product produces a scalar (number), while projection is an operation that produces a vector. I see what you're trying to say, however -- when the vector projected onto is a unit vector, its length is 1 and "disappears" from the denominators above.

- Warren

 

[AKG]

I believe that is correct for the scalar projection of U on V. I suppose you can say that the dot product is not exactly a projection, but a lot of the time you'll see the scalar projection of U on V given as U*V/|V|, where * represents the dot product operation.

 

[Gokul43201]

Or, the dot product is the product of the magnitude of the projection and the magnitude of the vector onto which the projection is made.

 

[gerben]

The dot produkt depends on the lengths of both vectors.
The dot produkt gives the length of the first vector times the lenth of the second vector times the cosine of the angle between them:
v1 dot v2 = Length(v1) * length(v2) * cos(angle)
so if in Zorodius question the length of V is 1 than his dot produkt gives him U cos(x)