Q$\mathrm{.}1.$ Given vectors $A=$hat$(x)2-$hat$(y)3+$hat$(z),B=$hat$(x)2-$hat$(y)+$hat$(z)3,$ and $C=$ hat$(x)2+$hat$(y)1-$hat$(z)1,$ show that $C$ is perpendicular to both A and $B.$

Q$\mathrm{.}2.$ Vector A starts at point $(1,-1,3)$ and ends at poin $(2.-1\mathrm{.}0).$ Find a unit vector in the direction of $A.$

Q$\mathrm{.}3.$ Given $A=$hat$(x)2-$hat$(y)3+$hat$(z)1$ and $B=$hat$(x)B_x+$hat$(y)2+$hat$(z)B_z$ :

$($a$)$ find $B_x$ and $B_z$ if $A$ is parallel to $B$;

$($b$)$ find a relation between $B_x$ and $B_z$ if $A$ is perpendicular to $B.$

Q$\mathrm{.}4.$ Given vectors $A=$hat$(x)+$hat$(y)2-$hat$(z)3,B=$hat$(x)2-$hat$(y)4,$ and $C=$ hat$(y)2-$hat$(z)4,$ find the following:

$($a$)$ A and $$

$($b$)$ The component of $B$ along $C$

$($c$){}_{AC}$

$($d$)AC$

$($e$)A*(BC)$

$($f$)A(BC)$

$($g$)$ hat$(x)B$

$($h$)(A$hat$(y))*hat(z)$

Q$\mathrm{.}5.$ Given $A=$hat$(x)(x+2y)-$hat$(y)(y+3z)+$hat$(z)(3x-y),$ determine a unit vector parallel to $A$ at point $P=(1,-1,2).$

Q$\mathrm{.}6.$ Given vectors $A=$hat$(x)2-$hat$(y)+$hat$(z)3$ and $B=$hat$(x)3-$hat$(z)2,$ find a vector $C$ whose magnitude is $9$ and whose direction is perpendicular to both A and $B.$