Dot vs Cross Product

24. Dot vs Cross Product

The dot product and cross product are the two fundamental products of vectors. They give complementary geometric information.

	Dot product $\vec{𝑎} \cdot \vec{𝑏}$	Cross product $\vec{𝑎} \times \vec{𝑏}$
Result type	scalar	vector (in $ℝ^{3}$ only)
Formula	$\vec{𝑎} \cdot \vec{𝑏} = \sum 𝑎_{𝑖} 𝑏_{𝑖}$	$[\begin{matrix} 𝑎_{2} 𝑏_{3} - 𝑎_{3} 𝑏_{2} \\ 𝑎_{3} 𝑏_{1} - 𝑎_{1} 𝑏_{3} \\ 𝑎_{1} 𝑏_{2} - 𝑎_{2} 𝑏_{1} \end{matrix}]$
Geometric formula	$\| \vec{𝑎} \| \| \vec{𝑏} \| cos 𝜃$	$\| \vec{𝑎} \| \| \vec{𝑏} \| sin 𝜃 \hat{𝑛}$
Geometric meaning	length of projection of $\vec{𝑎}$ onto $\vec{𝑏}$ , scaled by $\| \vec{𝑏} \|$	vector perpendicular to both, magnitude = parallelogram area
Zero when	vectors are perpendicular	vectors are parallel
Maximum when	vectors are parallel	vectors are perpendicular
Commutative?	yes: $𝑎 \cdot 𝑏 = 𝑏 \cdot 𝑎$	no: $𝑎 \times 𝑏 = - (𝑏 \times 𝑎)$

Combining the two formulas:

cos 𝜃 = \frac{\vec{𝑎} \cdot \vec{𝑏}}{‖ \vec{𝑎} ‖ ‖ \vec{𝑏} ‖} sin 𝜃 = \frac{‖ \vec{𝑎} \times \vec{𝑏} ‖}{‖ \vec{𝑎} ‖ ‖ \vec{𝑏} ‖}

𝜃 = arctan (‖ \vec{𝑎} \times \vec{𝑏} \frac{‖}{\vec{𝑎} \cdot \vec{𝑏}})

(This atan2-style form is numerically more stable than $arccos$ alone, especially near $𝜃 = 0$ or $𝜋$ .)

‖ \vec{𝑎} \times \vec{𝑏} ‖^{2} + {(\vec{𝑎} \cdot \vec{𝑏})}^{2} = ‖ \vec{𝑎} ‖^{2} ‖ \vec{𝑏} ‖^{2}

(Direct from ${sin}^{2} + {cos}^{2} = 1$ .)