Worked example 3.1: Broken play

Question: Major Applewhite receives the snap at the line of scrimmage, takes a seven step drop (i.e., runs backwards 9 yards), but is then flushed out of the pocket by a blitzing linebacker. Major subsequently runs parallel to the line of scrimmage for 12 yards and then gets off a forward pass, 36 yards straight downfield, to Roy Williams, just prior to being creamed by the linebacker. What is the magnitude of the football's resultant displacement (in yards)?
 
Answer: As illustrated in the diagram, the resultant displacement ${\bf r}$ of the football is the sum of vectors ${\bf a}$, ${\bf b}$, and ${\bf c}$, which correspond to the seven step drop, the run parallel to the line of scrimmage, and the forward pass, respectively.

Using the coordinate system indicated in the diagram, the components of vectors ${\bf a}$, ${\bf b}$, and ${\bf c}$ (measured in yards) are

$${\bf a} = (-9,0),$$

$${\bf b} = (0,12),$$

$${\bf c} = (36, 0),$$

respectively. Hence the components of ${\bf r}$ are given by

$${\bf r} = (x,y)= (-9+0+36, 0+12+0) = (27, 12).$$

It follows that the magnitude of the football's resultant displacement is

$$r = \sqrt{x^2+y^2} = \sqrt{27^2+12^2} = 29.55 {\rm yd}.$$