There are two forms of vector multiplication: one results
in a scalar, and one results in a vector.
The dot product, also called the scalar product,
takes two vectors, “multiplies” them together, and produces a scalar.
The smaller the angle between the two vectors, the greater their
dot product will be. A common example of the dot product in action
is the formula for work, which you will encounter in Chapter 4.
Work is a scalar quantity, but it is measured by the magnitude of
force and displacement, both vector quantities, and the degree to
which the force and displacement are parallel to one another.
The dot product of any two vectors, A
is expressed by the equation:
the angle made by A and B when they are placed tail to tail.
The dot product of A
the value you would get by multiplying the magnitude of A
the magnitude of the component of B
runs parallel to A
at the figure above, you can get A
multiplying the magnitude of A
the magnitude of
, which equals
You would get the same result if you multiplied the magnitude of B
the magnitude of
, which equals
Note that the dot product of two identical vectors is
their magnitude squared, and that the dot product of two perpendicular
vectors is zero.
the hands on a clock are vectors, where the hour hand has a length
of 2 and the minute hand has a length of 4. What is the dot product
of these two vectors when the clock reads 2 o’clock?
The angle between the hour hand and the minute
hand at 2 o’clock is 60º. With this information, we can simply plug
the numbers we have into the formula for the dot product:
The Cross Product
The cross product, also called the vector
product, “multiplies” two vectors together to produce a third vector,
which is perpendicular to both of the original vectors. The closer
the angle between the two vectors is to the perpendicular, the greater
the cross product will be. We encounter the cross product a great
deal in our discussions of magnetic fields. Magnetic force acts
perpendicular both to the magnetic field that produces the force,
and to the charged particles experiencing the force.
The cross product can be a bit tricky, because you have
to think in three dimensions. The cross product of two vectors, A
is defined by the equation:
is a unit vector perpendicular
to both A
The magnitude of the cross product vector is equal to the area made
by a parallelogram of A
In other words, the greater the area of the parallelogram, the longer
the cross product vector.
The Right-Hand Rule
You may have noticed an ambiguity here. The two vectors A and B always
lie on a common plane and there are two directions perpendicular
to this plane: “up” and “down.”
There is no real reason why we should choose the “up”
or the “down” direction as the right one, but it’s important that
we remain consistent. To that end, everybody follows the convention
known as the right-hand rule
. In order to find the
: Place the two
vectors so their tails are at the same point. Align your right hand
along the first vector, A
such that the base of your palm is at the tail of the vector, and
your fingertips are pointing toward the tip. Then curl your fingers
via the small angle toward the second vector, B
is in a clockwise
direction from A
find you have to flip your hand over to make this work. The direction
in which your thumb is pointing is the direction of
and the direction of
Note that you curl your fingers from A
the cross product is
. If it were written
you would have to curl your fingers from B
and your thumb would point downward. The order in which you write
the two terms of a cross product matters a great deal.
If you are right-handed, be careful! While you are working
hard on SAT II Physics, you may be tempted to use your left hand
instead of your right hand to calculate a cross product. Don’t do
once again that the minute hand of a clock is a vector of magnitude
4 and the hour hand is a vector of magnitude 2. If, at 5 o’clock,
one were to take the cross product of the minute handthe hour hand, what would the resultant
First of all, let’s calculate the magnitude of the cross
product vector. The angle between the hour hand and the minute hand
Using the right-hand rule, you’ll find that, by curling
the fingers of your right hand from 12 o’clock toward 5
o’clock, your thumb points in toward the clock. So the resultant vector
has a magnitude of 4 and points into the clock.