Question

Two points in the Cartesian plane are A(2.00 m, −4.00 m) and B(−3.00 m, 3.00 m). Find the distance between them and their polar coordinates.

Answer 1

The distance between the two points is
`d=8.6m`

The polar coordinate of A is
`\left(4.47,296.57\right)`

The polar coordinate of B is
`\left(4.24,135\right)`

Step-by-step explanation:

The two points are
`A(2,-4)` and
`B(-3,3)`

The distance between two points is given by,

`d=\sqrt{(2+3)^(2)+(-4-3)^(2)}\\d=\sqrt{(5)^(2)+(-7)^(2)}\\d=√(25+49)\\d=8.6`

Thus, the distance between the two points is
`d=8.6m`

The polar coordinates of A can be written as
`(Distance, tan^(-1) (y)/(x) )`

Distance =
`\sqrt{x^(2) +y^(2) }`

Substituting
`A(2,-4)`, we get,

Distance =
`\sqrt{2^(2) +(-4)^(2) }=√(4+16 )=4.47`

`tan^(-1) (y)/(x) =tan^(-1) (-4)/(2)=-63.43`

To make the angle positive, let us add 360,

`\theta=360-63.43=296.57`

The polar coordinate of A is
`\left(4.47,296.57\right)`

Similarly, The polar coordinate of B can be written as
`(Distance, tan^(-1) (y)/(x) )`

Distance =
`\sqrt{x^(2) +y^(2) }`

Substituting
`B(-3,3)`, we get,

Distance =
`\sqrt{(3)^(2)+(3)^(2)}=4.24`

`tan^(-1) (y)/(x) =tan^(-1) (3)/(-3)=-45`

To make the angle positive, let us add 360,

`\theta=180-45=135^(\circ)`

The polar coordinate of B is
`\left(4.24,135\right)`