Can the component of b in a direction with vectors be negative?

[burt]

The scalar projection of b in the a direction is a length. Can this be a negative value?

 

[Dr.Peterson]

Yes:

Scalar projection - Wikipedia

en.wikipedia.org

The scalar projection is a scalar, equal to the length of the orthogonal projection of [MATH]\mathbf {a}[/MATH] on [MATH]\mathbf {b}[/MATH], with a negative sign if the projection has an opposite direction with respect to [MATH]\mathbf {b}[/MATH].​

Note that it is not actually a length, but a signed length.

 

[pka]

The scalar projection of b in the a direction is a length. Can this be a negative value?

What is posted seems a bit confused.
The idea of projection evolves two vectors. You gave only one.
Look very carefully at the link provided by Prof Peterson.
Yes the direction of \(\displaystyle \vec{b}\) onto \(\displaystyle \vec{a}\), \(\displaystyle \frac{\vec{b}\cdot\vec{a}}{\vec{a}\cdot\vec{a}}\vec{a}\) .
That can be negative if the angle between the vectors is obtuse.

 

[burt]

You gave only one.

I gave two - the first mentioned is b and the second is a. I'm sorry - I don't know how to do vectors in mathjax so it's not clear.
(But I just looked at yours and now I do - \(\vec{b}\))

 

[Dr.Peterson]

An alternative way to represent vectors is just to use bold: a, b.