between points in a plane do not change when a coordinate system is rotated In other words, the magnitude of a vector Is invariant under rotations of the coordinate system. Suppose a coordinate system S is rotated about its origin by angle φ to become a new coordinate system S’ , as shown in the following figure. A point in a plane has coordinates ( x , y ) in S and coordinates ( x ' , y ' ) in S’. (a) Show that, during the transformation of rotation, the coordinates in S are expressed in terms of the coordinates in S by the following relations: { x ' = x cos φ + y sin φ y ' = − x sin φ + y cos φ (b) Show that the distance of point P to the origin is invariant under rotations of the coordinate system. Here, you have to show that x 2 + y 2 = x ' 2 + y ' 2 . (c) Show that the distance between points P and Q is invariant under rotations of the coordinate system. Here, you have to show that ( x P = x Q ) 2 + ( y P − y Q ) 2 = ( x ' P = x ' Q ) 2 + ( y ' P − y ' Q ) 2 .
between points in a plane do not change when a coordinate system is rotated In other words, the magnitude of a vector Is invariant under rotations of the coordinate system. Suppose a coordinate system S is rotated about its origin by angle φ to become a new coordinate system S’ , as shown in the following figure. A point in a plane has coordinates ( x , y ) in S and coordinates ( x ' , y ' ) in S’. (a) Show that, during the transformation of rotation, the coordinates in S are expressed in terms of the coordinates in S by the following relations: { x ' = x cos φ + y sin φ y ' = − x sin φ + y cos φ (b) Show that the distance of point P to the origin is invariant under rotations of the coordinate system. Here, you have to show that x 2 + y 2 = x ' 2 + y ' 2 . (c) Show that the distance between points P and Q is invariant under rotations of the coordinate system. Here, you have to show that ( x P = x Q ) 2 + ( y P − y Q ) 2 = ( x ' P = x ' Q ) 2 + ( y ' P − y ' Q ) 2 .
between points in a plane do not change when a coordinate system is rotated In other words, the magnitude of a vector Is invariant under rotations of the coordinate system. Suppose a coordinate system S is rotated about its origin by angle
φ
to become a new coordinate system S’, as shown in the following figure. A point in a plane has coordinates
(
x
,
y
)
in S and coordinates
(
x
'
,
y
'
)
in S’.
(a) Show that, during the transformation of rotation, the coordinates in S are expressed in terms of the coordinates in S by the following relations:
{
x
'
=
x
cos
φ
+
y
sin
φ
y
'
=
−
x
sin
φ
+
y
cos
φ
(b) Show that the distance of point
P
to the origin is invariant under rotations of the coordinate system. Here, you have to show that
x
2
+
y
2
=
x
'
2
+
y
'
2
.
(c) Show that the distance between points
P
and
Q
is invariant under rotations of the coordinate system. Here, you have to show that
(
x
P
=
x
Q
)
2
+
(
y
P
−
y
Q
)
2
=
(
x
'
P
=
x
'
Q
)
2
+
(
y
'
P
−
y
'
Q
)
2
.
Let vectors: A=(1,0,−3) B=(−2,5,1) and C=(3,1,1).
Calculate A×(B×C)
Calculate A⋅(B×C)
Calculate (2B)×(3C)
Given the points T (5, 20°, 120°) to P (2, 80°, 30°):
1.) find the unit vector in cartesian coordinates at T that is directedtoward P.
2.) find the unit vector in spherical coordinates at T that is directedtoward P.
The reason we typically use rectangular coordinates for two-dimensional or three-dimensional motion is that all the axes are orthogonal to each other. That is if we have unit vectors x^,y^,z^, then:
(1) x^⋅y^=x^⋅z^=y^⋅z^=0
So as long as your coordinate system uses axes that are all orthogonal to each other, you can use any coordinate system!For example, we can use the following coordinate systems:
POLAR COORDINATES f(r,θ):
Ax=rcosθ
Ay=rsinθ
Explain how Equation (1) applies to polar coordinates?
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