
Write each combination of directed line
segments as a single directed line segment.
 $\overrightarrow{PQ}+\overrightarrow{QR}$
 $\overrightarrow{RP}+\overrightarrow{PS}$
 $\overrightarrow{QS}\overrightarrow{PS}$
 $\overrightarrow{RS}+\overrightarrow{SP}+\overrightarrow{PQ}$

Given $A\left(2,3\right)$ and $B\left(2,1\right)$, find a vector $\mathbf{a}$ with representation given by
the directed line segment $\overrightarrow{AB}$. Draw $\overrightarrow{AB}$
and the equivalent representation starting at the
origin.
$\mathbf{a}=\u27e84,2\u27e9$

Find the sum of vectors $\u27e80,1,2\u27e9$ and
$\u27e80,0,3\u27e9$ and illustrate geometrically.
$\u27e80,1,1\u27e9$

Find $\mathbf{a}+\mathbf{b}$,
$2\mathbf{a}+3\mathbf{b}$,
$\left\mathbf{a}\right$,
and $\left\mathbf{a}\mathbf{b}\right$,
where
$\mathbf{a}=2\mathbf{i}4\mathbf{j}+4\mathbf{k}$
and
$\mathbf{b}=2\mathbf{j}\mathbf{k}$.

Find a unit vector with the same direction as
$8\mathbf{i}\mathbf{j}+4\mathbf{k}$.
$\frac{8}{9}\mathbf{i}\frac{1}{9}\mathbf{j}+\frac{4}{9}\mathbf{k}$

The two forces ${\mathbf{F}}_{1}$ and ${\mathbf{F}}_{2}$ with magnitudes $10$ lb and
$12$ lb respectively, act on an object at a point $P$ as
shown in the figure. Find the resultant force $\mathbf{F}$ acting at
$P$ as well as its magnitude and its direction. (Indicate
the direction by finding the angle $\theta $ shown in the
figure.)
$\mathbf{F}=\left(6\sqrt{3}5\sqrt{2}\right)\mathbf{i}+\left(6+5\sqrt{2}\right)\mathbf{j}$,
$\left\mathbf{F}\right\approx 13.5$ lb,
$\theta \approx 76\xb0$.

Velocities have both direction and
magnitude and thus are represented by vectors. The magnitude
of a velocity is
called speed. Suppose that a wind
is blowing from the direction N$45\xb0$W at a speed of $50$
km/h. (This means that the direction from which the wind
blows is $45\xb0$ west of northerly direction.) A pilot is
steering a helicopter in the direction N$60\xb0$E at an airspeed
(speed in still air) of $80$
km/h. The true course,
or track, of an aircraft is the
direction of the resultant of the velocity vectors of the
aircraft and the wind. The ground speed of the aircraft is the
magnitude of the resultant. Find the true course and the
ground speed of the helicopter.

Chains
$3$ and
$5$ m in length are
fastened to a suspended decoration. The decoration has a mass
of
$5$ kg. The chains, fastened at different heights, make
angles of
$52\xb0$ and
$40\xb0$ with the horizontal. Find
the tension in each chain and the magnitude of each
tension.

The tension at each end of the rope has
magnitude
$25$ N. What is the weight of the rope?
Let ${\mathbf{T}}_{1}$ and ${\mathbf{T}}_{2}$ be the tensions. By symmetry, their
horizontal components cancel and their vertical components
are equal. The magnitudes of vertical components
of ${\mathbf{T}}_{1}$ and ${\mathbf{T}}_{2}$ add up to $2\left{\mathbf{T}}_{1}\right\mathrm{sin}\left(37\xb0\right)$. Since the chain is at rest, the force ${\mathbf{T}}_{1}+{\mathbf{T}}_{2}$ must have the same magnitude as that of the force due
to gravity, $\leftm\mathbf{g}\right$, where $\left\mathbf{g}\right=9.8$. So we can
write
$2\left{\mathbf{T}}_{1}\right\mathrm{sin}\left(37\xb0\right)=9.8m$, and hence
$m=\frac{50\mathrm{sin}\left(37\xb0\right)}{9.8}$,
$m\approx 3.07$ [kg].

Use vectors to prove that the line joining
the midpoints of two sides of a triangle is parallel to the third side
and half its length.