Introduction on solving vector Cartesian coordinates:
This article is about solving vector Cartesian coordinates. Solving vector Cartesian coordinates is very simple. The tutors of tutor vista helps the students in solving vector Cartesian coordinate any time. The vector coordinates are x,y and z. Three simple methods are there to describe a vector. Directions, specific lengths, angles and projections or components are used to describe vectors. The simplest method of these is Cartesian or rectangular co-ordinate system. Below we can see about solving vector Cartesian coordinates.
Solving Vector Cartesian Coordinates
In cartesian coordinate system three co-ordinate axes x,y and z are mutually right angles to each other. Consider a point P(x,y,z) in space at a distance r from the orgin. The vector r can be represented as
r = `xbara_x``+ y bara_y + z bara_z`
Where `bara_x, bara_y,` and `bara_z` are unit vectors
x,y and z are the components vectors. Components vectors have a magnitude and direction. Unit vectors have unit magnitude and directed along the co-ordinate axis.
A unit vector in a given direction is a vector in that direction divided by its magnitude. It is given by
`a_r = r/|r|`
`a_r = (xbara_x + ybara_y + z bara_z)/sqrt (x^2 + y^2 + z^2)`
Consider the points P(x,y,z)and Q(x + dx, y +dy, z+dz) in rectangul;ar co-ordinate system. The differential length dl from P to Q is the diagonal of the parallel piped is given by
dl = sqrt ((dx)^2 + (dy)^2 + (dz)^2)
The differential area ds = dx dy
= dy dz
= dz dx
The differential volume dv = dx dy dz
Solving Vector Cartesian Coordinates
Conversion of cylindrical to cartesian system
The cylinderical co-ordinates (?, F, z) can be converted into cartesian co-ordinates ( x, y, z)
Given Transform
? x = r cos ?
f y = r sin f
z z = z
conversion of spherical to cartesian co-ordinates system
The spherical co-ordinates (r, `theta` , f) can be transformed into cartesian co-ordinates (x, y, z)
Given Transform
r x = r sin ?. cosf
? y = r sin ? sin f
f z =r cos ?
Solving Vector Cartesian Coordinates
Example Problem: Give the Cartesian co-ordinates of a point whose cylinderical are ? = 1, f = 45°, z =2.
Given
Cylinderical co-ordinates are ? = 1, f = 45°, z =2.
Cartesian co-ordinates are x, y, z
x = ? cosf
= 1.cos 45 = 0.707
y = ? sinf
= 1. sin 45 = 0.707
z = z
= 2
Cartesian co-ordinates are (0.707, 0.707, 2)
This article is about solving vector Cartesian coordinates. Solving vector Cartesian coordinates is very simple. The tutors of tutor vista helps the students in solving vector Cartesian coordinate any time. The vector coordinates are x,y and z. Three simple methods are there to describe a vector. Directions, specific lengths, angles and projections or components are used to describe vectors. The simplest method of these is Cartesian or rectangular co-ordinate system. Below we can see about solving vector Cartesian coordinates.
Solving Vector Cartesian Coordinates
In cartesian coordinate system three co-ordinate axes x,y and z are mutually right angles to each other. Consider a point P(x,y,z) in space at a distance r from the orgin. The vector r can be represented as
r = `xbara_x``+ y bara_y + z bara_z`
Where `bara_x, bara_y,` and `bara_z` are unit vectors
x,y and z are the components vectors. Components vectors have a magnitude and direction. Unit vectors have unit magnitude and directed along the co-ordinate axis.
A unit vector in a given direction is a vector in that direction divided by its magnitude. It is given by
`a_r = r/|r|`
`a_r = (xbara_x + ybara_y + z bara_z)/sqrt (x^2 + y^2 + z^2)`
Consider the points P(x,y,z)and Q(x + dx, y +dy, z+dz) in rectangul;ar co-ordinate system. The differential length dl from P to Q is the diagonal of the parallel piped is given by
dl = sqrt ((dx)^2 + (dy)^2 + (dz)^2)
The differential area ds = dx dy
= dy dz
= dz dx
The differential volume dv = dx dy dz
Solving Vector Cartesian Coordinates
Conversion of cylindrical to cartesian system
The cylinderical co-ordinates (?, F, z) can be converted into cartesian co-ordinates ( x, y, z)
Given Transform
? x = r cos ?
f y = r sin f
z z = z
conversion of spherical to cartesian co-ordinates system
The spherical co-ordinates (r, `theta` , f) can be transformed into cartesian co-ordinates (x, y, z)
Given Transform
r x = r sin ?. cosf
? y = r sin ? sin f
f z =r cos ?
Solving Vector Cartesian Coordinates
Example Problem: Give the Cartesian co-ordinates of a point whose cylinderical are ? = 1, f = 45°, z =2.
Given
Cylinderical co-ordinates are ? = 1, f = 45°, z =2.
Cartesian co-ordinates are x, y, z
x = ? cosf
= 1.cos 45 = 0.707
y = ? sinf
= 1. sin 45 = 0.707
z = z
= 2
Cartesian co-ordinates are (0.707, 0.707, 2)
