# Dot product with a unit vector in the direction of $b$: what does it represent?

Let $$F:\mathbb{R}^3\to\mathbb{R}^3$$ be of class $$C^2$$ and let $$a, b\in\mathbb{R}^3$$ be two nonnull vectors. Let $$F(x) = F(x_1, x_2, x_3)$$.

Consider the vector given by $$\left(\sum_{i=1}^3 \frac{\partial F_1}{\partial x_i} a_i, \sum_{i=1}^3 \frac{\partial F_2}{\partial x_i} a_i, \sum_{i=1}^3 \frac{\partial F_3}{\partial x_i} a_i \right).$$

Denoting by $$\widehat{b}$$ the unit vector in the direction of the vector $$b$$, I am trying to understand what $$\left(\sum_{i=1}^3 \frac{\partial F_1}{\partial x_i} a_i, \sum_{i=1}^3 \frac{\partial F_2}{\partial x_i} a_i, \sum_{i=1}^3 \frac{\partial F_3}{\partial x_i} a_i \right)\cdot\widehat{b}$$ does represent.

If, e.g. $$b$$ is a vector in the direction of the $$y$$-axis, I would say that $$\left(\sum_{i=1}^3 \frac{\partial F_1}{\partial x_i} a_i, \sum_{i=1}^3 \frac{\partial F_2}{\partial x_i} a_i, \sum_{i=1}^3 \frac{\partial F_3}{\partial x_i} a_i \right)\cdot\widehat{b} = \sum_{i=1}^3 \frac{\partial F_2}{\partial x_i} a_i,$$ but I am not sure what does that scalar product represent in general (I mean, $$b$$ could be in an "easy" direction, but also in the $$x+y$$ direction and others more complicated).

You can take $$\hat b$$ inside the derivative:

$$\left(\sum_{i=1}^3 \frac{\partial F_1}{\partial x_i} a_i, \sum_{i=1}^3 \frac{\partial F_2}{\partial x_i} a_i, \sum_{i=1}^3 \frac{\partial F_3}{\partial x_i} a_i \right)\cdot\widehat{b}=\sum_{i=1}^3\frac{\partial(F\cdot\hat b)}{\partial x_i}a_i\;.$$

So this is the directional derivative along $$a$$ of the scalar function $$F\cdot\hat b$$, the component of $$F$$ along $$\hat b$$.