We can use vectors to solve many problems involving physical quantities such as velocity, speed, weight, work and so on.

The velocity of moving object is modeled by a vector whose direction is the direction of motion and whose magnitude is the speed.

**Example : **

A ball is thrown with an initial velocity of $70$ feet per second., at an angle of $35\xb0$ with the horizontal. Find the vertical and horizontal components of the velocity.

Let $\text{v}$ represent the velocity and use the given information to write $\text{v}$ in unit vector form:

$\text{v}\text{\hspace{0.17em}}=\text{70}\left(\mathrm{cos}(35\xb0)\right)i+70\left(\mathrm{sin}(35\xb0)\right)j$

Simplify the scalars, we get:

$\text{v}\text{\hspace{0.17em}}\approx 57.34i+40.15j$

Since the scalars are the horizontal and vertical components of $\text{v}$,

Therefore, the horizontal component is $57.34$ feet per second and the vertical component is $40.15$ feet per second.

Force is also represented by vector. If several forces are acting on an object, the resultant force experienced by the object is the vector sum of these forces.

**Example : **

Two forces ${\text{F}}_{1}$ and ${\text{F}}_{2}$ with magnitudes $20$ and $30\text{\hspace{0.17em}}\text{lb}$, respectively, act on an object at a point $\text{P}$ as shown. Find the resultant forces acting at $\text{P}$.

First we write ${\text{F}}_{1}$ and ${\text{F}}_{2}$ in component form:

$\text{v}\approx 57.34i+40.15j$

Simplify the scalars, we get:

$\begin{array}{l}\begin{array}{l}{\text{F}}_{1}=\left(20\mathrm{cos}\left(45\xb0\right)\right)i+\left(20\mathrm{sin}\left(45\xb0\right)\right)j\\ =20\left(\frac{\sqrt{2}}{2}\right)i+20\left(\frac{\sqrt{2}}{2}\right)j\end{array}\hfill \\ =10\sqrt{2}i+10\sqrt{2}j\hfill \\ \begin{array}{l}{\text{F}}_{2}=\left(30\mathrm{cos}\left(150\xb0\right)\right)i+\left(30\mathrm{sin}\left(150\xb0\right)\right)j\\ =30\left(-\frac{\sqrt{3}}{2}\right)i+30\left(\frac{1}{2}\right)j\end{array}\hfill \\ =-15\sqrt{3}i+15j\hfill \end{array}$

So, the resultant force $\text{F}$ is

$\begin{array}{l}\text{F}={\text{F}}_{1}+{\text{F}}_{2}\\ =\left(10\sqrt{2}\text{\hspace{0.17em}}i+10\sqrt{2}j\right)+\left(-15\sqrt{3}\text{\hspace{0.17em}}i+15j\right)\\ =\left(10\sqrt{2}-15\sqrt{3}\right)i+\left(10\sqrt{2}+15\right)j\\ \approx -12i+29j\end{array}$

The work $\text{W}$ done by a force $\text{F}$ in moving along a vector $\text{D}$ is $\text{W}=\text{F}\cdot \text{D}$.

**Example : **

A force is given by the vector $\text{F}=\langle 2,3\rangle $ and moves an object from the point $(1,3)$ to the point $(5,9)$. Find the work done.

First we find the Displacement.

The displacement vector is

$\text{D}=\langle 5-1,9-3\rangle =\langle 4,6\rangle $.

By using the formula, the work done is

$\text{W}=\text{F}\cdot \text{D}=\langle 2,3\rangle \cdot \langle 4,6\rangle =26$

If the unit of force is pounds and the distance is measured in feet, then the work done is $26$ ft-lb.