Mathematics

Problem

Non-zero vectors

\vec p

and

\vec q

satisfy

|\vec p| = |\vec q| = 5

and

\vec p \cdot \vec q = 0

. a. (i) Prove that

\bigl|(\vec p + \vec q) \times (\vec p - \vec q)\bigr| = 50

. b. (ii) Hence find the area of the parallelogram whose adjacent sides are

\vec p + \vec q

and

\vec p - \vec q

. c. (iii) Compare the area obtained in part (ii) with the sum of the areas of the squares having sides

\vec p

and

\vec q

, and interpret the result geometrically.