Inequalities with vectors

Inequalities with vectors involve comparing the lengths (magnitudes) of vectors, or the results of operations like the dot product. These inequalities help us understand geometric relationships, such as angles between vectors or distances in the plane.

Important properties

• The triangle inequality: For any vectors $\vec{a}$ and $\vec{b}$, $|\vec{a} + \vec{b}| \leq |\vec{a}| + |\vec{b}|$.

• The reverse triangle inequality: $|\,|\vec{a}| - |\vec{b}|\,| \leq |\vec{a} - \vec{b}|$.

• Cauchy-Schwarz inequality: $|\vec{a} \cdot \vec{b}| \leq |\vec{a}|\,|\vec{b}|$.

• Equality in these inequalities usually occurs when the vectors are parallel or have special relationships.