The dot product of the two vectors in cartesian coordinates is defined as: Multiply corresponding components and the add the results. Thus
\( \vec a \cdot \vec b=a_1b_1+a_2b_2+a_3b_3\)
If there are two vectors in cylindrical coordinates,
\(\vec{A} = A_r \hat{r} + A_\theta \hat{\theta} + A_z \hat{z}\)
\(\vec{B} = B_r \hat{r} + B_\theta \hat{\theta} + B_z \hat{z}\)
What’s the dot product? Is it \(\vec{A} \bullet \vec{B} = (A_rB_r + A_\theta B_\theta + A_z B_z)\)?
According to the definition of dot product, it is the product of the lengths of the vectors and the cosine of the angle between them. So the dot product may be:
\(\vec{A} \bullet \vec{B} = (A_rB_r \cos (A_\theta- B_\theta) + A_z B_z)\)
Let’s transform the two vectors into the cartesian coordinates and calulate the dot products
\(\vec{A} = A_r \cos A_\theta \hat{\text{i}} + A_r \sin A_\theta \hat{\text{j}} + A_z \hat{\text{k}}\)
\(\vec{B} = B_r \cos B_\theta \hat{\text{i}} + B_r \sin B_\theta \hat{\text{j}} + B_z \hat{\text{k}}\)
\(\vec{A} \bullet \vec{B} = A_r B_r \cos A_\theta \cos B_\theta + A_r B_r \sin A_\theta \sin B_\theta + A_z B_z\)
Using the triangle identity \(\cos (x-y)=\cos x \cos y+\sin x\sin y\) to simplify,
\(\vec{A} \bullet \vec{B} = A_r B_r \cos (A_\theta – B_\theta) + A_z B_z\)