Mathematics

Problem

Given the vectors

\mathbf a=(k,1,2)

\mathbf b=(1,k,-1)

and

\mathbf c=(2,-1,k)

. a. (i) Write the two independent conditions that must be satisfied for

\mathbf b

to be perpendicular to both

\mathbf a

and

\mathbf c

. b. (ii) Solve the simultaneous conditions to find all real values of

k

. c. (iii) For each value of

k

obtained, decide whether the angle between

\mathbf a

and

\mathbf c

is acute, obtuse or right-angled, giving justification in each case.