Questions on direction of dot product

Question

So i am having problems with understanding that if two forces (suppose in parallelogram law of addition of two vector) the resultant has the same direction as the diagonal between them. So theta (angle between resultant and the base vector) represents the direction of resultant by that cos theta shouldn't have hand in showing the direction. Am I right? Second question is that in dot products how can the product of two vector be scalar? Because from the image as you can see the vector is moving in the forward direction, then how it is scalar?image shows a body is pulled by a force f and cover displacement s

Answer 1

Welcome! Angle $\theta$ is the angle between the force vector $\vec{F}$ and the displacement vector (toward the moving direction) $\vec{r}$.

So the work done by the force is $W=\vec{F}\cdot \vec{r}$

The middle $\cdot$ is called the dot-product, and it is defined as:

$$\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos(\theta)$$

where $\theta$ is the angle between the two vector, and $|\vec{a}|$ is the length of the vector $\vec{a}$. So the length is a non-negative number, and $\cos(\theta)$ is a number between $-1$ and $1$, so the dot-product gives a number, and we call it "scalar".

Now, go back to the work, $$W=\vec{F}\cdot \vec{r}=|\vec{F}||\vec{r}|\cos(\theta)$$