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
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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 hand the 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
