#include

#include

using namespace std;

const double EPSILON$(1$e$-11)$;

$//$ function prototypes

$//$ ENTER FUNCTION PROTOTYPE FOR read$\_$vector HERE.

$//$ ENTER FUNCTION PROTOTYPE FOR vec$\_$length HERE.

$//$ ENTER FUNCTION PROTOTYPE FOR dot$\_$product HERE.

$//$ ENTER FUNCTION PROTOTYPE FOR compute$\_$angle HERE.

$//$ ENTER FUNCTION PROTOTYPE FOR radians$\_$to$\_$degrees HERE.

$//$ ENTER FUNCTION PROTOTYPE FOR normalization HERE.

$//$ ENTER FUNCTION PROTOTYPE FOR output$\_$vector HERE.

$//$ ENTER FUNCTION PROTOTYPE FOR resultant$\_$vector HERE

$//$ ENTER FUNCTION PROTOTYPE FOR display$\_$info HERE.

$//***$ DO NOT CHANGE ANY CODE IN THE MAIN FUNCTION.

int main$()$

$\{$

double u$1,$ v$1$; $//$ coordinates of vector $1$

double u$2,$ v$2$; $//$ coordinates of vector $2$

double radians; $//$ angle in radians

double degrees; $//$ angle in degrees

read$\_$vector$("$Enter $1$st vector $(2$ floats$)$: $",$ u$1,$ v$1)$;

read$\_$vector$("$Enter $2$nd vector $(2$ floats$)$: $",$ u$2,$ v$2)$;

$//$ compute angle in radians between $($u$1,$ v$1)$ and $($u$2,$ v$2)$

radians $=$ compute$\_$angle$($u$1,$ v$1,$ u$2,$ v$2)$;

$//$ convert radians to degrees

degrees $=$ radians$\_$to$\_$degrees$($radians$)$;

$//$ output information

display$\_$info$($u$1,$ v$1,$ u$2,$ v$2,$ degrees$)$;

cout $<<$ endl;

return$(0)$;

$\}$

$//$ DEFINE FUNCTION read$\_$vector HERE.

$//$ DEFINE FUNCTION vec$\_$length HERE.

$//$ DEFINE FUNCTION dot$\_$product HERE.

$//$ DEFINE FUNCTION compute$\_$angle HERE.

$//$ DEFINE FUNCTION radians$\_$to$\_$degrees HERE.

$//$ DEFINE FUNCTION normalization HERE.

$//$ DEFINE FUNCTION output$\_$vector HERE.

$//$ DEFINE FUNCTION resultant$\_$vector HERE

$//$ DEFINE FUNCTION display$\_$info HERE.

Test $5$

$>$ run

Enter $1$st vector $(2$ floats$)$: $1\mathrm{.}25-0\mathrm{.}25$

Enter $2$nd vector $(2$ floats$)$: $0\mathrm{.}353\mathrm{.}14$

First vector: $(1\mathrm{.}25,-0\mathrm{.}25)$ has length $1\mathrm{.}27475$

Second vector: $(0\mathrm{.}35,3\mathrm{.}14)$ has length $3\mathrm{.}15945$

Resultant vector: $(1\mathrm{.}6,2\mathrm{.}89)$ has length $3\mathrm{.}30335$

Angle between vectors $(1\mathrm{.}25,-0\mathrm{.}25)$ and $(0\mathrm{.}35,3\mathrm{.}14)=94\mathrm{.}9497$ degrees.

The vectors are NOT ORTHOGONAL.

Test $6$

$>$ run

Enter $1$st vector $(2$ floats$)$: $0\mathrm{.}353\mathrm{.}14$

Enter $2$nd vector $(2$ floats$)$: $1\mathrm{.}25-0\mathrm{.}25$

First vector: $(0\mathrm{.}35,3\mathrm{.}14)$ has length $3\mathrm{.}15945$

Second vector: $(1\mathrm{.}25,-0\mathrm{.}25)$ has length $1\mathrm{.}27475$

Resultant vector: $(1\mathrm{.}6,2\mathrm{.}89)$ has length $3\mathrm{.}30335$

Angle between vectors $(0\mathrm{.}35,3\mathrm{.}14)$ and $(1\mathrm{.}25,-0\mathrm{.}25)=94\mathrm{.}9497$ degrees.

The vectors are NOT ORTHOGONAL$\mathrm{.}78$

Test $7$

$>$ run

Enter $1$st vector $(2$ floats$)$: $0\mathrm{.}707106781180\mathrm{.}70710678118$

Enter $2$nd vector $(2$ floats$)$: $-0\mathrm{.}707106781180\mathrm{.}70710678118$

First vector: $(0\mathrm{.}707107,0\mathrm{.}707107)$ has length $1$

Second vector: $(-0\mathrm{.}707107,0\mathrm{.}707107)$ has length $1$

Resultant vector: $(0,1\mathrm{.}41421)$ has length $1\mathrm{.}41421$

Angle between vectors $(0\mathrm{.}707107,0\mathrm{.}707107)$ and $(-0\mathrm{.}707107,0\mathrm{.}707107)=90$ degrees.

The vectors are ORTHONORMAL.

Test $8$

$>$ run

Enter $1$st vector $(2$ floats$)$: $-0\mathrm{.}707106781180\mathrm{.}70710678118$

Enter $2$nd vector $(2$ floats$)$: $-0\mathrm{.}70710678118-0\mathrm{.}70710678118$

First vector: $(-0\mathrm{.}707107,0\mathrm{.}707107)$ has length $1$

Second vector: $(-0\mathrm{.}707107,-0\mathrm{.}707107)$ has length $1$

Resultant vector: $(-1\mathrm{.}41421,0)$ has length $1\mathrm{.}41421$

Angle between vectors $(-0\mathrm{.}707107,0\mathrm{.}707107)$ and $(-0\mathrm{.}707107,-0\mathrm{.}707107)=90$ degrees.

The vectors are ORTHONORMAL.

Test $9$

$>$ run

Enter $1$st vector $(2$ floats$)$: $0-5$

Enter $2$nd vector $(2$ floats$)$: $-10$

First vector: $(0,-5)$ has length $5$

Second vector: $(-1,0)$ has length $1$

Resultant vector: $(-1,-5)$ has length $5\mathrm{.}09902$

Angle between vectors $(0,-5)$ and $(-1,0)=90$ degrees.

The vectors are ORTHOGONAL, but NOT ORTHONORMAL.