Mathematics

Problem

Let

\vec a

and

\vec b

be any two non-zero vectors such that

\vec a\cdot\vec b = 0

and

|\vec a| = |\vec b|

. a. (i) Prove that the vectors

\vec a + \vec b

and

\vec a - \vec b

are perpendicular to each other. b. (ii) For

\displaystyle \vec a = 3\hat i + 4\hat j

and

\displaystyle \vec b = -4\hat i + 3\hat j

, verify numerically the result proved in part (i) by evaluating a suitable product. c. (iii) Show that

|\vec a + \vec b| = |\vec a - \vec b|

under the given conditions and explain the geometrical significance of this equality with reference to a square constructed on the vectors.