Chapter 9.CR, Problem 1CC

a) What is a vector in the plane? How do we represent a vector in the coordinate plane?

b) Find the vector with initial point $\left(2,3\right)$ and terminal point $\left(4,10\right)$.

c) Let $v=\langle 2,1\rangle$. If the initial point $v$ is placed at $P\left(1,1\right)$, where is its terminal point? Sketch several representations of $v$.

d) How is the magnitude of $v=\langle a_1,a_2\rangle$ defined? Find the magnitude of $w=\langle 3,4\rangle$.

e) What are the vectors $i$ and $j$? Express the vector $v=\langle 5,9\rangle$ in terms of $i$ and $j$.

f) Let $v=\langle a_1,a_2\rangle$ be a vector in the coordinate plane. What is meant by the direction $\theta$ of $v$? What are the coordinates of $v$ in terms of its length and direction? Sketch a figure to illustrate your answer.

g) Suppose that $v$ has length $\left|v\right|=5$ and direction $\theta =\pi /6$. Express $v$ in terms of its coordinates.