Question $2(20$ Marks$)$

$($a$)$ If the position of vectors of point $T=3a_x-2a_y-a_z$ and point

$S=4a_x+6a_y+2a_z,$ find:

i$.$ The coordinates of $T$ and $S.$

$(2$ marks$)$

ii$.$ The distance vector from $T$ to $S.$

$(2$ marks$)$

iii. The distance between $T$ and $S.$

$(2$ marks$)$

$($b$)$ If $A=5a_x+3a_y-2a_z,B=-a_x+4a_y+6a_z,$ and $C=8a_x+2a_y,$

$(8$ marks$)$

find the values of $$ and $$ such that $A+B+C$ is parallel to

the $y-$axis.

$($c$)$ Show that $(A*B{)}^2+(AB{)}^2=(AB{)}^2.$

$(6$ marks$)$