This article is from the In-Depth Report The Science of Pro Football

What Are Vectors, and How Are They Used?

Denoting both direction and magnitude, vectors appear throughout the world of science and engineering

In the Vectors episode of NBC Learn's "The Science of NFL Football" you see that quarterbacks must account for their own motion when throwing a pass, and that both the player's movement and the path of the ball can be represented by arrows known as vectors.

Vectors are used in science to describe anything that has both a direction and a magnitude. They are usually drawn as pointed arrows, the length of which represents the vector's magnitude. A quarterback's pass is a good example, because it has a direction (usually somewhere downfield) and a magnitude (how hard the ball is thrown).

Off the field, vectors can be used to represent any number of physical objects or phenomena. Wind, for instance, is a vectorial quantity, because at any given location it has a direction (such as northeast) and a magnitude (say, 45 kilometers per hour). You could make a map of airflow at any point in time, then, by drawing wind vectors for a number of different geographic locations.

Many properties of moving objects are also vectors. Take, for instance, a billiard ball rolling across a table. The ball's velocity vector describes its movement—the direction of the vector arrow marks the ball's direction of motion, and the length of the vector represents the speed of the ball.

The billiard ball's momentum is also a vectorial quantity, because momentum is equal to mass times velocity. Therefore, the ball's momentum vector points in the same direction as its velocity vector, and the momentum vector's magnitude, or length, is the multiplication product of the ball's speed and its mass.

Momentum vectors are useful when you want to predict what will happen when two objects come into contact. Recall from the video that vectors can be added together by joining them to make a shape called a parallelogram and finding the diagonal of that parallelogram. The diagonal is the sum of the two vectors that form the sides of the parallelogram.

Let's say that a rolling billiard ball is moving toward a glancing collision with a stationary billiard ball. On impact, the moving ball transfers some of its momentum to the stationary ball, and both roll away from the collision in different directions. Following the impact, both balls have velocity and hence momentum. In fact, the sum of the momentum vectors of the two balls after the collision is equal to the first ball's momentum vector before the collision, ignoring small losses due to friction as well as sound and heat energy produced during the impact.

So, with an understanding of vectors, billiards players can predict where both balls will go following a collision, allowing them to sink more target balls while keeping the cue ball safely on the table.

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