Finding a Force using the dot product/projection (Calc 3)

[amalia]

Homework Statement

This is from Larson's Calculus Early Transcendentals 4th Ed (Pg. 786)

A 600-pound boat sits on a ramp imclined at 30 degrees, as shown in Figure 11.32. What force is required to keep the boat from rolling down the ramp?

Homework Equations

The solution gives the following equation: w₁ = proj_vF = [(F * v) / ||v||²] = (F * v)v

The Attempt at a Solution

This is just an example in the book, and I'm confused about how the author did the last two steps in the previous equation. I think it's just one of those omissions where it's really simple, but I can't figure out why the length of the vector squared can just disappear.

 

[diazona]

The second to last step makes sense, since that's just the vector projection formula:
$$\frac{(\vec{u}\cdot\vec{v})\vec{v}}{v^2}$$
is the projection of $\vec{u}$ on to $\vec{v}$. But I'm not understanding the last step. The norm squared can't just disappear like that, unless it's a unit vector (in which case the norm is 1).

 

[amalia]

Someone just told me it is a unit vector, but thanks for your help anyway! I had been wondering if that could just "cancel" or or not...