cement $(\backslash (\backslash$vec$\{$d$\}\backslash ))**$: The shortest straight$-$line distance between two points, along with a specified direction. For example, $5$ km east.

$2.**$Velocity $(\backslash (\backslash$vec$\{$v$\}\backslash ))**$: The rate of change of displacement over time, with direction. For instance, $30$ m$/$s north.

$3.**$Force $(\backslash (\backslash$vec$\{$F$\}\backslash ))**$: A push or pull on an object, represented by both its strength $($e$.$g$.,10$ N$)$ and the direction $($e$.$g$.,$ upward$).$

$4.**$Acceleration $(\backslash (\backslash$vec$\{$a$\}\backslash ))**$: The rate of change of velocity, with direction.

##### Characteristics of Vectors:

$-$ Vectors require direction for their complete description.

$-$ Vectors are added or subtracted using vector algebra, such as the $**$triangle law$**$ or $**$parallelogram law$**.$

##### Representation:

Vectors are written with an arrow over the symbol, e$.$g$.,\backslash (\backslash$vec$\{$F$\}\backslash ),$ or as bold letters, e$.$g$.,\backslash (\backslash$mathbf$\{$F$\}\backslash ).$

$---$

#### $**$Key Differences Between Scalars and Vectors$**$

$|$ Property $|$ Scalar $|$ Vector $|$

$|--------------------|-------------------------|----------------------------|$

$|**$Definition$**|$ Only magnitude $|$ Magnitude and direction $|$

$|**$Representation$**|$ Single numerical value $|$ Arrow with magnitude and direction $|$

$|**$Examples$**|$ Distance, Speed, Time $|$ Displacement, Velocity, Force $|$

$|**$Addition Rule$**|$ Simple arithmetic $|$ Vector addition $($e$.$g$.,$ triangle or parallelogram rule$)|$

$---$

### $**$Mathematical Formulas for Scalars and Vectors$**$

$1.**$Speed$**(\backslash ($s$\backslash ))$:

$\backslash [$

s $=\backslash$frac$\{\backslash$text$\{$Distance$\}\}\{\backslash$text$\{$Time$\}\}=\backslash$frac$\{$d$\}\{$t$\}$

$\backslash ]$

$2.**$Velocity$**(\backslash (\backslash$vec$\{$v$\}\backslash ))$:

$\backslash [$

$\backslash$vec$\{$v$\}=\backslash$frac$\{\backslash$text$\{$Displacement$\}\}\{\backslash$text$\{$Time$\}\}=\backslash$frac$\{\backslash$vec$\{$d$\}\}\{$t$\}$

$\backslash ]$

$3.**$Acceleration$**(\backslash (\backslash$vec$\{$a$\}\backslash ))$:

$\backslash [$

$\backslash$vec$\{$a$\}=\backslash$frac$\{\backslash$Delta $\backslash$vec$\{$v$\}\}\{\backslash$Delta t$\}$

$\backslash ]$

where $\backslash (\backslash$Delta $\backslash$vec$\{$v$\}\backslash )$ is the change in velocity, and $\backslash (\backslash$Delta t $\backslash )$ is the time interval.

$4.**$Resultant Vector Magnitude$**$:

For two vectors at right angles:

$\backslash [$

R $=\backslash$sqrt$\{$x$^2+$ y$^2\}$

$\backslash ]$

where $\backslash ($x$\backslash )$ and $\backslash ($y$\backslash )$ are the perpendicular components of the vector.

$5.**$Resultant Vector Direction$**$:

$\backslash [$

$\backslash$theta $=\backslash$tan$^\{-1\}\backslash$left$(\backslash$frac$\{\backslash$text$\{$Opposite Component$\}\}\{\backslash$text$\{$Adjacent Component$\}\}\backslash$right$)$

$\backslash ]$

$---$

### $**$Worked Examples$**$

#### $**$Example $1$: Distance vs$.$ Displacement$**$

$**$Problem$**$: A person walks $3$ km east and then $4$ km west. What is their distance and displacement?

$**$Solution$**$:

$-**$Distance$**$:

$\backslash [$

$\backslash$text$\{$Distance$\}=3\backslash ,\backslash$text$\{$km$\}+4\backslash ,\backslash$text$\{$km$\}=7\backslash ,\backslash$text$\{$km$\}$

$\backslash ]$

$-**$Displacement$**$:

Displacement is the shortest distance between the start and end points, with direction.

$\backslash [$

$\backslash$text$\{$Displacement$\}=4\backslash ,\backslash$text$\{$km west$\}-3\backslash ,\backslash$text$\{$km east$\}=1\backslash ,\backslash$text$\{$km west$\}$

$\backslash ]$

$---$

#### $**$Example $2$: Speed vs$.$ Velocity$**$

$**$Problem$**$: A car travels $120$ km north in $2$ hours. Calculate the speed and velocity.

$**$Solution$**$:

$-**$Speed$**$:

$\backslash [$

s $=\backslash$frac$\{\backslash$text$\{$Distance$\}\}\{\backslash$text$\{$Time$\}\}=\backslash$frac$\{120\backslash ,\backslash$text$\{$km$\}\}\{2\backslash ,\backslash$text$\{$hours$\}\}=60\backslash ,\backslash$text$\{$km$/$h$\}$

$\backslash ]$

$($Speed is a scalar, so no direction is needed.$)$

$-**$Velocity$**$:

$\backslash [$

$\backslash$vec$\{$v$\}=\backslash$frac$\{\backslash$text$\{$Displacement$\}\}\{\backslash$text$\{$Time$\}\}=\backslash$frac$\{120\backslash ,\backslash$text$\{$km north$\}\}\{2\backslash ,\backslash$text$\{$hours$\}\}=60\backslash ,\backslash$text$\{$km$/$h north$\}$

$\backslash ]$

$($Velocity is a vector, so direction is specified.$)$

$---$

#### $**$Example $3$: Resultant Displacement$**$

$**$Problem$**$: A person walks $3$ km north and then $4$ km east. Find the resultant displacement and direction.

$**$Solution$**$:

$-$ The displacement forms a right triangle with the two paths as perpendicular sides.

$-**$Magnitude$**$:

$\backslash [$

R $=\backslash$sqrt$\{(3^2+4^2)\}=\backslash$sqrt$\{9+16\}=\backslash$sqrt$\{25\}=5\backslash ,\backslash$text$\{$km$\}$

$\backslash ]$

$-**$Direction$**$:

$\backslash$