Vector Projection: Is it Possible for projuv=projvu?

[IniquiTrance]

Homework Statement

Is it possible for

proj_uv=proj_vu

Homework Equations

The Attempt at a Solution

This can only occur if:

$$\frac{|\mathbf{u\cdot v}|}{^{\|u\|^{2}}}\mathbf{u} = \frac{|\mathbf{u\cdot v}|}{^{\|v\|^{2}}}\mathbf{v}$$

So if either is the zero vector, it is true. How can I prove that it can only be true if either is the zero vector, or v=u?

 

[Office_Shredder]

You can immediately see from

$$\frac{|\mathbf{u\cdot v}|}{^{\|u\|^{2}}}\mathbf{u} = \frac{|\mathbf{u\cdot v}|}{^{\|v\|^{2}}}\mathbf{v}$$

That u is a multiple of v (and vice versa). What do you think beyond that?

 

[IniquiTrance]

You can immediately see from



That u is a multiple of v (and vice versa). What do you think beyond that?

That if one is larger/smaller than the other, the projections cannot be equivalent. Is that sufficient to prove it?

 

[Office_Shredder]

That if one is larger/smaller than the other, the projections cannot be equivalent.

Do you have a reason for believing that?

If u is parallel to v, then

$$\frac{|\mathbf{u\cdot v}|}{^{\|u\|^{2}}}\mathbf{u}$$

is parallel to
$$\frac{|\mathbf{u\cdot v}|}{^{\|v\|^{2}}}\mathbf{v}$$

So how can we tell whether they are equal?

 

[IniquiTrance]

Is it because:

$$\frac{\mathbf{u}}{^{\|u\|}}\frac{1}{\|u\|} = \frac{\mathbf{v}}{^{\|v\|}}\frac{1}{\|v\|}$$

So we have a constant times a unit vector on each side, therefore the constants must be equal?