Dot product in the plane

Given a pair of 2D vectors:

$\overrightarrow{u}=(u_x,u_y)$

$\overrightarrow{v}=(v_x,v_y)$

Then their dot product is

$\overrightarrow{u}\cdot \overrightarrow{v}:=u_x{v}_x+u_y{v}_y$.

Suppose that $\overrightarrow{u}$ and $\overrightarrow{v}$ are both rotated by an angle $\theta$:

$\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)\left(\begin{array}{c}{u}_x\\ u_y\end{array}\right)=\left(\begin{array}{c}{u}_x\cos \theta -u_y\sin \theta \\ u_x\sin \theta +u_y\cos \theta \end{array}\right)=\left(\begin{array}{c}{u_x}^{\prime }\\ {u_y}^{\prime }\end{array}\right)$

$\{\begin{array}{c}{u_x}^{\prime }=u_x\cos \theta -u_y\sin \theta \\ {u_y}^{\prime }=u_x\sin \theta +u_y\cos \theta \end{array}$

$\{\begin{array}{c}{v_x}^{\prime }=v_x\cos \theta -v_y\sin \theta \\ {v_y}^{\prime }=v_x\sin \theta +v_y\cos \theta \end{array}$

${\overrightarrow{u}}^{\prime }\cdot {\overrightarrow{v}}^{\prime }={u_x}^{\prime }{v_x}^{\prime }+{u_y}^{\prime }{v_y}^{\prime }$

$=(u_x\cos \theta -u_y\sin \theta )(v_x\cos \theta -v_y\sin \theta )+(u_x\sin \theta +u_y\cos \theta )(v_x\sin \theta +v_y\cos \theta )$

$=u_x{v}_x{\cos }^2\theta -u_x{v}_y\cos \theta \sin \theta -u_y{v}_x\sin \theta \cos \theta +u_y{v}_y{\sin }^2\theta$

$+u_x{v}_x{\sin }^2\theta +u_x{v}_y\sin \theta \cos \theta +u_y{v}_x\cos \theta \sin \theta +u_y{v}_y{\cos }^2\theta$

$=u_x{v}_x({\cos }^2\theta +{\sin }^2\theta )+u_y{v}_y({\cos }^2+{\sin }^2\theta )$

$=u_x{v}_x+u_y{v}_y=\overrightarrow{u}\cdot \overrightarrow{v}$

applying the trigonometric form of the Pythagorean Theorem (i.e., ${\cos }^2\theta +{\sin }^2\theta =1$).

So if $\overrightarrow{u}$ and $\overrightarrow{v}$ are both rotated with the angle between them preserved then their dot product is preserved as well.

Multiplying lengths of $\overrightarrow{u}$ and $\overrightarrow{v}$. Let ${\overrightarrow{u}}^{\prime }=\lambda \overrightarrow{u}$, ${\overrightarrow{v}}^{\prime }=\mu \overrightarrow{v}$. I.e.,

${\overrightarrow{u}}^{\prime }=(\lambda u_x,\lambda u_y)$,

${\overrightarrow{v}}^{\prime }=(\mu v_x,\mu v_y)$.

Then

${\overrightarrow{u}}^{\prime }\cdot {\overrightarrow{v}}^{\prime }=\lambda \mu u_x{v}_x+\lambda \mu u_y{v}_y$

$=\lambda \mu (u_x{v}_x+u_y{v}_y)=\lambda \mu \overrightarrow{u}\cdot \overrightarrow{v}$.

Multiplying the length of $\overrightarrow{u}$ or $\overrightarrow{v}$ also multiplies the length of the dot product $\overrightarrow{u}\cdot \overrightarrow{v}$ by the same factor. (This property is called homogeneity, or linearity.)

What happens when $\hat{u}$ and $\hat{v}$ are unit vectors? Rotate both of them (by the same angle) until one of them equals the vector (1, 0). Suppose that ${\hat{u}}^{\prime }=(1,0)$. Then ${\hat{v}}^{\prime }=({v_x}^{\prime },{v_y}^{\prime })$,

${\hat{u}}^{\prime }\cdot {\hat{v}}^{\prime }={v_x}^{\prime }$.

Let $\varphi$ be the angle between ${\hat{u}}^{\prime }$ and ${\hat{v}}^{\prime }$ (or between $\hat{u}$ and $\hat{v}$, equivalently).

$\cos \varphi =\frac{{v_x}^{\prime }}{1}={v_x}^{\prime }$

so ${\hat{u}}^{\prime }\cdot {\hat{v}}^{\prime }=\cos \varphi =\hat{u}\cdot \hat{v}$

For any vectors $\overrightarrow{u},\overrightarrow{v}$ which are not unit vectors, let

$\overrightarrow{u}=u\hat{u}$,

$\overrightarrow{v}=v\hat{v}$

where $u=\sqrt{\overrightarrow{u}\cdot \overrightarrow{u}}$ is the magnitude (or length) of vector $\overrightarrow{u}$ (by the Pythagorean Theorem) and $v=\sqrt{\overrightarrow{v}\cdot \overrightarrow{v}}$ likewise, and where $\hat{u}$ is a unit vector in the same direction as $\overrightarrow{u}$; and $\hat{v}$ is a unit vector in the same direction as $\overrightarrow{v}$.

So

$\overrightarrow{u}\cdot \overrightarrow{v}=u\hat{u}\cdot v\hat{v}=uv(\hat{u}\cdot \hat{v})$

$\overrightarrow{u}\cdot \overrightarrow{v}=uv\cos \varphi$

where $\varphi$ is the angle between $\hat{u}$ and $\hat{v}$ (or equivalently, between $\overrightarrow{u}$ and $\overrightarrow{v}$). This last equation denotes the geometric interpretation of the dot product (as being proportional to the magnitudes of the vectors and the cosine of the angle between them).