Mathematics

Problem

Consider the vectors

\mathbf{i},\;\mathbf{j},\;\mathbf{k}

of the three–dimensional Cartesian coordinate system. a. (i) State the algebraic condition that two vectors

\mathbf{a}

and

\mathbf{b}

must satisfy in order to be perpendicular to each other. b. (ii) Without performing any numerical calculation, explain why

\mathbf{i},\;\mathbf{j}

and

\mathbf{k}

are pairwise perpendicular. c. (iii) Let

\mathbf{u}=a\mathbf{i}+b\mathbf{j}+c\mathbf{k}

. Give a geometric argument to show that the vector

\mathbf{v}=b\mathbf{i}-a\mathbf{j}

is perpendicular to

\mathbf{u}

whenever

c=0

. (You may use the condition stated in part (i) but should not perform a component–wise dot product.)