Changes

Its Cartesian components are:

Its Cartesian components are:

::<math>(A_x=- r \sin \theta, A_y=r \cos \theta)\,</math>

::<math>(A_x=- r \sin \theta, A_y=r \cos \theta)\,</math>

It always points "laterally", at right angles to the radial direction.  More surprisingly, its length is <math>\sqrt{A_x^2 + A_y^2} = r\,</math>.  The angular basis vector is ''not'' a unit vector!  It is bigger when farther away from the origin, even though the sum of the squares of its components is 1.<ref>Some textbooks use a definition of vector components such that basis vectors always have length 1.  Such a treatment imposes the requirement that the coordinate lines be perpendicular everywhere.  The formulas for vector operations are more complicated and less elegant when this is done.</ref>  In general curvilinear coordinates, the "square root of the sum of the squares of the components" rule is not correct for the length of a vector.  The correct rule will be given [[some link to the norm|below]].

It always points "laterally", at right angles to the radial direction.  More surprisingly, its length is <math>\sqrt{A_x^2 + A_y^2} = r\,</math>.  The angular basis vector is ''not'' a unit vector!  It is bigger when farther away from the origin, even though the sum of the squares of its components is 1.<ref>Some textbooks use a definition of vector components such that basis vectors always have length 1.  Such a treatment imposes the requirement that the coordinate lines be perpendicular everywhere.  The formulas for vector operations are more complicated and less elegant when this is done.</ref>  In general curvilinear coordinates, the "square root of the sum of the squares of the components" rule is not correct for the length of a vector.  The correct rule will be given [[Calc3.2#The_Length_of_a_Vector_in_Arbitrary_Coordinates|below]].

===The Dot Product in Arbitrary Coordinates===

===The Dot Product in Arbitrary Coordinates===

===The Length of a Vector in Arbitrary Coordinates===

===The Length of a Vector in Arbitrary Coordinates===

The length of a vector (sometimes called the "norm") is the square root of the dot product of the matrix with itself.  Note that this is defined in terms of something that we have defined in a purely geometrical way, independent of any coordinate system.  Note also that it is consisten with our earlier definition of the dot product of two vectors as the product of their lengths, times the cosine of the angle between them.

The length of a vector (sometimes called the "norm") is the square root of the dot product of the matrix with itself.  Note that this is defined in terms of something that we have defined in a purely geometrical way, independent of any coordinate system.  Note also that it is consistent with our earlier definition of the dot product of two vectors as the product of their lengths, times the cosine of the angle between them.

The length of a vector is usually written with a sort of double absolute value sign.

The length of a vector is usually written with a sort of double absolute value sign.