Problem $2$

Even though visualizations are useful for us to gain intuitions, in practice it's convenient to decompose vectors into components and compute the algebra with their components. The vector hat$(x)$ denotes a unit vector $($i$.$e$.$ a vector whose length is one$)$ that points in the positive $x-$direction, and similarly for hat$(y).$ With this notation, for example, one can think of a vector vec$(V)=5$hat$(x)+3$hat$(y)$ as literally a vector $5$hat$(x)$ plus another vector $3$hat$(y).$ When expressing the vector vec$(V)$ in this fashion, we sometimes say that we express the vector vec$(V)$ in the component form. Equivalently, one may also denote vec$(V)$ as vec$(V)=(5,3).$

$($a$)$ Draw the vector vec$(V)$ in the grid below.

$($b$)$ Based on the Pythagorean theorem, find the magnitude $|vec(V)|$ of the vector vec$(V).($The magnitude of a vector is its length.$)$