NHL players are celebrated for their ability to pass the puck quickly and accurately as play moves from one end of the ice to the other. These pinpoint passes, requiring both magnitude and direction, are perfect examples of velocity vectors. "Science of NHL Hockey" is a 10-part video series produced in partnership with the National Science Foundation and the National Hockey League.

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LESTER HOLT, reporting: In the National Hockey League, players are celebrated for their skating, their checking, and their scoring. But one of the most overlooked skills is the art of passing.

MATT MOULSON (Left Wing, New York Islanders): I think passing is, you know, probably one of the more important things in hockey today. I don't think I’d have too many goals if I didn't play with guys who are great passers.

HOLT: To get the puck safely to a teammate's stick takes speed, accuracy, and incredible vision.

BRENDEN MORROW (Left Wing, Dallas Stars): If a player is a good passer, he can see a play develop and read that and kind of be a quarterback and lead the player with the pass.

HOLT: It also takes something in math and physics called a "velocity vector." A moving puck is a great example of a velocity vector because it has both a speed and direction. The velocity vector of the puck is represented with an arrow, with the head representing its direction and the length representing its speed.

Dr. EDWARD BURGER (Williams College, Baylor University): Depending upon the length of the vector, that will tell you how fast it will get to player B. If it's a long vector it will get there really fast. If it's a short vector it will take a long time to get there. But that's a very simple example where you can see the vector of the puck going to player B.

HOLT: But making the perfect pass isn't so simple in a real hockey game - because the puck isn't the only thing moving.

BURGER: There are multiple vectors at play, of course, because if you’re watching a game in any one moment everything is moving. The players are moving. The sticks are moving. And of course, the puck is moving. And so whenever there's movement, there's a vector associated with it.

HOLT: Players must be aware of their speed and position on the ice and also the speed and eventual position of their teammates.

ERIK JOHNSON (Defenseman, Colorado Avalanche): The effort is putting it where you want to put it before the guy you are passing to gets there. So if the guy is cutting across the ice, I'm not passing where he is, I'm passing where he's going to be.

HOLT: Although NHL players figure this out intuitively, there are two ways to add vectors to find the vector the puck needs to move from point-A to point-B. The first way is called the "head-to-tail" method.

Imagine this is the velocity vector of the puck at point-A, and the player wants the puck to end up here at point-B. He makes the pass with this velocity vector. But the puck will also continue in this direction.

BURGER: And now to find the sum of this vector and this vector, I can just literally go from the tail of the first to the head of the second and you can imagine I'm creating a triangle.

HOLT: The vector of the puck and the vector of the pass are placed head-to-tail to find their sum - a third vector, called the resultant - which is the actual velocity the puck will travel to reach the target.

BURGER: And that ending point defines the ending point of my new vector which starts at the original spot here, from here to here. Boom.

HOLT: The second way to add vectors is done with something called the "parallelogram method." This method uses a parallelogram, a four-sided object in which the opposite sides are parallel.

Dr. IRENE FONSECA (Carnegie Melon University): So I have my vector A and I have my vector B. So I take the vector B so that they both have the same tail point. And then I draw a parallelogram. So two sides parallel to [vector] A, two sides parallel to vector B. The diagonal of that parallelogram is the resultant. It's exactly the same vector obtained with the first method.

HOLT: An NHL player is calculating all this with lightning-speed while he skates and scrambles and passes the puck.

MOULSON: It's a split second decision that you have to make and know where to put a puck for your teammate, for the best possible chance for them to make a play with it.

BURGER: But, of course, that's what's exciting about hockey. It's the game of chaos. But vectors are behind the scene turning the chaos into mathematics.

HOLT: Velocity vectors: the invisible physics behind the perfect hockey pass.

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