Angle Between Two Vectors

Use $\cos\theta = \dfrac{\vec a\cdot\vec b}{|\vec a||\vec b|}$, then $\theta = \cos^{-1}(\ldots)$. For $\vec a = (1, 0)$ and $\vec b = (1, 1)$, $\cos\theta = 1/\sqrt 2$, so $\theta = 45^\circ$.

Enter components of $\vec a$ and $\vec b$. The lab returns the dot product, magnitudes and the angle in radians and degrees. The principal value lies in $[0, \pi]$. Try $\vec a = (1, 0, 0)$, $\vec b = (0, 1, 0)$ for $\theta = 90^\circ$.

Dot product gives $\cos\theta$ — fast for orthogonality and angle magnitude. Cross product gives $|\vec a\times\vec b| = |\vec a||\vec b|\sin\theta$ — useful when you also need a perpendicular vector. Both yield $\theta$, but dot is simpler when only angle matters.

Solar panel tilt: power generation is proportional to $\cos\theta$ between sunlight direction and panel normal. Wind turbines, antennas pointing at satellites, and even cricket bat-ball collision angles can be analysed this way. The dot product is your angle-meter.