Let a and b be two non-zero vectors. Prove that | a × b | ≤| a || b |. State the condition under which equality holds, i.e., | a × b |=| a || b |.

$$|\vec{\mathrm{a}} \times \vec{\mathrm{b}}|=|\vec{\mathrm{a}}||\vec{\mathrm{b}}||\sin \theta| $$ As, $\displaystyle 0 \leq|\sin \theta| \leq 1$ $\displaystyle \left.\Rightarrow\right|^{\vec{\mathrm{a}}}| | \vec{\mathrm{b}}| | \sin \theta\left|\leq\left.\right|^{\vec{\mathrm{a}}}\right||\vec{\mathrm{b}}|$ $\displaystyle \Rightarrow|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}| \leq|\overrightarrow{\mathrm{a}}||\overrightarrow{\mathrm{b}}|$ For equality, $\displaystyle \sin \theta=1 \Rightarrow \theta=\frac{\pi}{2} \Rightarrow \vec{\mathrm{a}}$ is perpendicular to $\displaystyle \vec{\mathrm{b}}$.