(Solved): PLEASE HELP!!! Answer all with clear steaps so i can understand. Will give thumbs up (1) Suppose ...

(1) Suppose $\Vert \vec{p}\Vert =3,\Vert \vec{q}\Vert =4$, and the angle between $\vec{p}$ and $\vec{q}$ is $\pi /3$. Find $\vec{p}\cdot \vec{q}$. (2) Suppose $\Vert \vec{p}\Vert =8,\Vert \vec{q}\Vert =3$, and $\vec{p}\cdot \vec{q}=-12$. Find the angle between $\vec{p}$ and $\vec{q}$ in radians. (3) Let $\vec{p}$ be a vector with $\Vert \vec{p}\Vert =5$. (a) What is the maximum value of $\vec{p}\cdot \vec{q}$, where $\vec{q}$ is allowed to be any unit vector? (b) What vector $\vec{q}$ achieves the maximum value from part (a)? (4) Let $\vec{u}=[3,2,-6]$ and $\vec{v}=[12,16,-5]$. (a) Find proj $\vec{u}\vec{v}$. (b) Find ${\mathrm{comp}}_{\vec{u}}\vec{v}$. (5) Suppose $\vec{v}$ and $\vec{v}$ are vectors in the $xy$-plane, each with initial point at the origin. The length of $\vec{u}$ is 4 , the length of $\vec{v}$ is 6 , and the angle from $\vec{v}$ to $\vec{v}$ is $5\pi /6$ in the counterclockwise direction. (a) What is the magnitude and direction of $\vec{u}\times \vec{v}$ ? (b) Find the components of $\vec{v}\times \vec{v}$. (6) Suppose $\vec{u}=\vec{i}+2\vec{j}$ and $\vec{v}=\vec{j}+\vec{k}$. Find $\vec{u}\times \vec{v}$.