# Homework Help: Vector projection

1. Sep 20, 2009

### IniquiTrance

1. The problem statement, all variables and given/known data

Is it possible for

projuv=projvu

2. Relevant equations

3. 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?

2. Sep 20, 2009

### Office_Shredder

Staff Emeritus
You can immediately see from

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

3. Sep 20, 2009

### IniquiTrance

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

4. Sep 20, 2009

### Office_Shredder

Staff Emeritus
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?

5. Sep 20, 2009

### 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?