MAT 21C Lecture Notes - Lecture 14: Lorentz Force, Cross Product, Parallelogram
MAT 21C – Lecture 14 – Vectors and the Cross Product
• The dot produt ultiplies to etors to gie a salar uer. If e hae
two vectors,
, then
(geometrical definition)
• The algebraic definition of the dot product is
. It
follows that
and
if and only if
and are orthogonal.
• The geometrical definition of the cross product: The normal vector,
has a
direction orthogonal to both
and , (
) where
in that order is a
right-handed system.
• The cross product of two vectors,
, is
. The
magnitude of the cross product is
.
• The angle from
is represented by .
• Example: Standard 3 dimensional unit vectors:
It follows that
Then
. If or
then define
. Vectors
are parallel if and only if
.
• Applications where the cross product arise in physical situations
o 1) Charged particle: A particle has a charge, q and velocity
in a magnetic
field
. Lorentz force =
.
o 2) Meteorology/Weather: The Corolis force =
• Example: Find the area of a parallelogram with sides:
.
The area of the parallelogram,
.
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MAT 21C Full Course Notes
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Mat 21c lecture 14 vectors and the cross product: the dot produ(cid:272)t (cid:862)(cid:373)ultiplies(cid:863) t(cid:449)o (cid:448)e(cid:272)tors to gi(cid:448)e a s(cid:272)alar (cid:894)(cid:374)u(cid:373)(cid:271)er(cid:895). If (cid:449)e ha(cid:448)e two vectors, (cid:1873) (cid:1853)(cid:1866)(cid:1856) (cid:1874) , then (cid:1873) (cid:1874) =|(cid:1873) ||(cid:1874) |cos (geometrical definition: the algebraic definition of the dot product is (cid:1873) (cid:1874) = (cid:1873)(cid:2869)(cid:1874)(cid:2869)+(cid:1873)(cid:2870)(cid:1874)(cid:2870)+(cid:1873)(cid:2871)(cid:1874)(cid:2871). The magnitude of the cross product is |(cid:1873) (cid:1876)(cid:1874) |=|(cid:1873) ||(cid:1874) ||sin|: the angle from (cid:1873) (cid:1872)(cid:1867) (cid:1874) is represented by , example: standard 3 dimensional unit vectors: (cid:2835) = ,(cid:2836) = (cid:1853)(cid:1866)(cid:1856) = . It follows that (cid:2835) (cid:1876)(cid:2836) =(cid:1855)| |,|(cid:2835) |=(cid:883),(cid:1853)(cid:1866)(cid:1856) |(cid:2836) |=(cid:883). ; (cid:2836) (cid:1876)(cid:2835) = ; (cid:2836) (cid:1876) =(cid:2835) ; (cid:1853)(cid:1866)(cid:1856) (cid:1876)(cid:2836) = (cid:2835) . If =(cid:882) or (cid:1873) ,(cid:1874) =(cid:882) then define (cid:1873) (cid:1876)(cid:1874) = (cid:882). Vectors (cid:1873) ,(cid:1874) are parallel if and only if (cid:1873) (cid:1876)(cid:1874) =(cid:882): 1) charged particle: a particle has a charge, q and velocity (cid:1873) in a magnetic field (cid:1828) .
