Dot product

Essential knowledge 3.2.A.3.i in the AP Physics C: Mechanics CED states that the dot product between two vectors results in a scalar quantity of magnitude $AB\cos\theta$, with $\theta$ the angle between them, in degrees. In components, $\vec{A}\cdot\vec{B} = A_xB_x + A_yB_y + A_zB_z$.

The sign follows the cosine, and that is most of its physical content:

• Aligned vectors, $\theta = 0$ degrees, give the largest positive value $AB$.
• Perpendicular vectors, $\theta = 90$ degrees, give exactly zero.
• Opposed vectors, $\theta = 180$ degrees, give $-AB$.

Work is the standard case. The C: Mechanics sheet prints $W = \int_a^b \vec{F}\cdot d\vec{r}$, and the algebra-based sheets print the constant-force version $W = F_{\parallel}d = Fd\cos\theta$. Same statement: only the component of force along the displacement transfers energy. A perpendicular force, such as the tension on a ball swung in a circle, does no work at all, and the zero at 90 degrees is why.

Flux is the other case. AP Physics 2 prints magnetic flux as $\Phi_B = \vec{B}\cdot\vec{A}$, and C: Electricity and Magnetism prints $\Phi_E = \int\vec{E}\cdot d\vec{A}$. Only the field component through the surface counts, so a surface turned edge on to the field carries no flux.

The result is a number, never an arrow.