202004231606 Exercise 1, Section 1.1

Determine whether the vectors emanating from the origin and terminating at the following pair of points are parallel.

(a) $(3,1,2)$ and $(6,4,2)$

(b) $(-3,1,7)$ and $(9,-3,-21)$

(c) $(5,-6,7)$ and $(-5,6,-7)$

(d) $(2,0,-5)$ and $(5,0,-2)$

Background.

Two nonzero vectors $x$ and $y$ are called parallel if $y=tx$ for some nonzero real number $t$ . (Thus nonzero vectors having the same or opposite directions are parallel.)

_{Text on pg.3}

Solution.

(a) Let $x=(3,1,2)$ and $y=(6,4,2)$ . Apparently $\nexists\, t\in \mathbb{R}$ such that $y=tx$ . For otherwise $(6,4,2)=t(3,1,2)$ , the system of equations

$\begin{aligned} 6 & = 3t \\ 4 & = 1t \\ 2 & = 2t \\ \end{aligned}$

is inconsistent. They are not parallel.

(b) Let $x=(-3,1,7)$ and $y=(9,-3,-21)$ , then $y=-3x$ . The vectors $x$ and $y$ are in opposite direction and the magnitude of $y$ is three times that of $x$ . They are parallel.

(c) Let $x=(5,-6,7)$ and $y=(-5,6,-7)$ . Observe that they are in equal magnitude but in opposite direction, i.e., $y=-x$ . They are also parallel.