Mathematics

Problem

Let

\mathbf{a}=\langle 3,-2,1\rangle

and

\mathbf{b}=\langle k,4,-2\rangle

, where

k

is a real number. a. (i) Calculate

\mathbf{a}\!\cdot\!\mathbf{b}

when

k=1

. b. (ii) Find the value of

k

for which

\mathbf{a}

and

\mathbf{b}

are perpendicular. c. (iii) For the value of

k

obtained in part (ii), determine the angle between

\mathbf{a}

and

\mathbf{b}+\mathbf{a}

, giving your answer in degrees to the nearest whole number.