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# Chapter 3 - Vectors

## Vectors and scalars

Vector quantities with number, direction and units:

• Displacement $\vec{r}$ [m]
• Velocity $\vec{v}$ [ms-1]
• Acceleration $\vec{a}$ [ms-2]

Scalar quantities number and units only

• Distance traveled [m]
• Speed [ms-1]

## Graphical representation of vectors and components

It is frequently useful to draw two dimensional vectors as arrows, and to split them in to components that lie along the coordinate axes. The choice of coordinate axes is up to you..but choosing the right ones will make the problem easier or harder.

We can take a look at the acceleration due to gravity as vector using a phone accelerometer, using this tool (click on the link from your phone's browser).

## Unit Vectors

It can be useful to express vector quantities in terms of unit vectors. These are dimensionless vectors of length = 1 that point along the coordinate axes. They are usually denoted with carets (hats), i.e. $(\hat{i},\hat{j},\hat{k})$

For example:

$\vec{v}\,\mathrm{ms^{-1}}=v_{x}\,\mathrm{ms^{-1}}\,\hat{i}+v_{y}\,\mathrm{ms^{-1}}\,\hat{j}+v_{z}\,\mathrm{ms^{-1}}\,\hat{k}$

or

$\vec{r}\,\mathrm{m}=x\,\mathrm{m}\,\hat{i}+y\,\mathrm{m}\,\hat{j}+z\,\mathrm{m}\,\hat{k}$

## Vectors and motion

$\vec{r_{1}}=x_{1}\,\hat{i}+y_{1}\,\hat{j}+z_{1}\,\hat{k}$

$\vec{r_{2}}=x_{2}\,\hat{i}+y_{2}\,\hat{j}+z_{2}\,\hat{k}$

$\Delta\vec{r}=\vec{r_{2}}-\vec{r_{1}}=(x_{2}-x_{1})\,\hat{i}+(y_{2}-y_{1})\,\hat{j}+(z_{2}-z_{1})\,\hat{k}$

Average velocity: $\vec{v_{ave}}=\frac{\Delta\vec{r}}{\Delta t}$

Instantaneous velocity: $\vec{v}=\frac{d\vec{r}}{dt}=\frac{dx}{dt}\,\hat{i}+\frac{dy}{dt}\,\hat{j}+\frac{dz}{dt}\,\hat{k}=v_{x}\,\hat{i}+v_{y}\,\hat{j}+v_{z}\,\hat{k}$

Average acceleration: $\vec{a_{ave}}=\frac{\Delta\vec{v}}{\Delta t}$

Instantaneous acceleration: $\vec{a}=\frac{d\vec{v}}{dt}=\frac{dv_x}{dt}\,\hat{i}+\frac{dv_y}{dt}\,\hat{j}+\frac{dv_z}{dt}\,\hat{k}=\frac{d^{2}x}{dt^2}\,\hat{i}+\frac{d^{2}y}{dt^2}\,\hat{j}+\frac{d^{2}z}{dt^2}\,\hat{k}$

## Vectors - Components

 $v_{1x}=v_{1}\cos\theta_{1}$ $v_{2x}=v_{2}\cos\theta_{2}$ $v_{1y}=v_{1}\sin\theta_{1}$ $v_{2y}=v_{2}\sin\theta_{2}$ $v_{Rx}=v_{1x}+v_{2x}$ $\tan\theta_{R}=\frac{v_{Ry}}{v_{Rx}}$ $v_{Ry}=v_{1y}+v_{2y}$ $v_{R}=\sqrt{v_{Rx}^2+v_{Ry}^2}$

## Vectors - Multiplication by a scalar

Multiplication of a vector by a scalar can change the magnitude, but not the direction of the vector, ie. each component of the vector is multiplied by the scalar in the same way.