If vec$(v)$ has magnitude $20$ and points at an angle $,$ measured counter$-$clockwise from the positive $x-$direction $($this is called a direction angle$),$ find a general formula for the components of vec$(v)$ in terms of the angle $.$

Calculate the magnitude and the direction angle $($in radians$)$ for the vector vec$(v)=($:$3,-3\sqrt[\phantom{2}]{3}$:$).$

Find a unit vector $($that$\text{'}$s a vector of magnitude $1)$ that points in the same direction as vec$(v)=($:$-6,8$:$).$

Find a vector vec$(v)($in component form$)$ so that $||vec(v)||=392$ and when drawn in standard position vec$(v)$ makes a $117$ angle with the positive $x-$axis.

Calculate the magnitude and direction angle $($in degrees$)$ for the vector vec$(v)=($:$-7,3$:$).$