Vector Multiplication
Vector Multiplication
There are two forms of vector multiplication: one results in a scalar, and one results in a vector.
Dot Product
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 and B, is expressed by the equation:
where is the angle made by A and B when they are placed tail to tail.
The dot product of A and B is the value you would get by multiplying the magnitude of A by the magnitude of the component of B that runs parallel to A. Looking at the figure above, you can get A · B by multiplying the magnitude of A by the magnitude of , which equals . You would get the same result if you multiplied the magnitude of B by 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.
Example
Suppose 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 and B, is defined by the equation:
where is a unit vector perpendicular to both A and B. The magnitude of the cross product vector is equal to the area made by a parallelogram of A and B. 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 cross product, : 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. If B is in a clockwise direction from A, you’ll 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 to B because the cross product is . If it were written , you would have to curl your fingers from B to A, 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 this.
Example
Suppose 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 vector be?
First of all, let’s calculate the magnitude of the cross product vector. The angle between the hour hand and the minute hand is 150º:
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.
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