11.2 Vectors In Space

This section is about the properties of vectors in 3-D space. The component (regular) form of a vector is
v = <v1,v2,v3>. As we know from earlier chapters, a vector is the difference between a terminal and initial point. Since this is in three dimensions, the three numbers inside the angle brackets are the differences between the x-values, y-values, and z-values, respectively, of a terminal and initial point.


Length of a vector: ||v|| = squareroot(v1^2 + v2^2 + v3^2)

• This equation makes sense, as the length of a vector would be the distance between its initial point and its terminal point. Each vn value is the difference between the x-, y-, or z-value of the terminal and initial points, so this length equation is the same as the basic distance formula we all know and love!

11.4 Lines and Planes In Space

In this lesson, we learned how to find the parametric and symmetric equations of a line in a 3-D space and how to find the angle and distance between two planes. In this blog, I'm going to focus on planes. The basic equation is:

a(x-x1) + b(y-y1) + b(z-z1) = 0

In this equation:

• a,b,c are the x, y, and z components of a perpendicular (normal) vector. n = <a,b,c>
• x1, y1, and z1 are the coordinates of a point on the plane.
• x,y, and z are variables. 

Many problems will give hints and ask you to find the equation of a plane. In the following example, a point on the plane and a perpendicular vector are given:

Cool 3D Shapes

The Importance of Steals

I read an interesting article about the importance of steals in the NBA last night. The writer, Benjamin Morris, measured the performance of teams with and without certain players. He compared the team's points per game with and without the player to the player's average stats and a replacement player's average stats; the chart below shows his results. For example, if a team is playing without a player that gets 12 rebounds per game who is being replaced by a player with just 2 rebounds per game, the team will probably average 17 fewer points.

As you can see in this chart, a steal is 9 times more valuable than a point! It's high value is understandable, as a steal will almost always result in two easy points in the resulting fast-break scoring opportunity and also shifts the momentum, but it's amazing that such an overlooked stat is so valuable. It's easier to glorify high-scoring players because the numbers are so much grander; Kevin Durant is leading the league with 32.2 points per game, while Chris Paul leads the league in steals with 2.53 steals per game. But let's look at the stats of these two players with the logic of this chart:

Durant:

32.2 PPG

1.3 STLPG

0.8 BLKPG

5.6 APG

7.6 RPG

3.6 TOPG

32.2(1) + 1.3(9.1) + 0.8(6.1) + 5.6(2.2) + 7.6(1.7) - 3.6(5.4) = 54.71

Chris Paul:

18.5 PPG

2.5 STLPG

0.1 BLKPG

11.0 APG

4.3 RPG

2.4 TOPG

18.5(1) + 2.5(9.1) + 0.1(6.1) + 11(2.2) + 4.3(1.7) - 2.4(5.4) = 60.41

With the logic of a player's contributions compared to a replacement player, it seems like our own Chris Paul of the Los Angeles Clippers is an even better player than Kevin Durant, the leading MVP candidate!

Works Cited:

http://espn.go.com/nba/player/_/id/2779/chris-paul

http://espn.go.com/nba/player/_/id/3202/kevin-durant

http://fivethirtyeight.com/features/the-hidden-value-of-the-nba-steal/

10.8 Polar Equations of Conics

In this section, we are finding the polar equations of conics and graphing them on polar graphs. In this example, we are given the type of conic (parabola), the eccentricity (1), and the directrix (y=-2). With a horizontal directrix below the origin, we can tell that the equation's denominator will be (1-esinØ). The directrix is also 2 units away from the focus [which is always at (0,0) for these problems], so we know that p is 2. The bottom row of this picture displays how to insert those numbers and find the final equation (boxed on the right).

10.6 Polar Coordinates

This section introduces us to graphing using polar coordinates rather than the usual rectangular (x,y) coordinates. Polar coordinates are (r,Ø) [that Ø is the closest thing to theta I could get]. The four equations in the picture below are used to transfer polar equations into rectangular equations and vice versa. 

This next picture is an example of changing a polar equation into a rectangular one. Knowing the angle, you can plug it into the top-left equation from the above picture. The final answer is boxed below:

This printable paper is helpful to use for graphing polar equations:

http://www.embeddedmath.com/downloads/files/polargraph/polargraph-letter.pdf

Wilt Chamberlain vs. The Modern NBA

In my free time, I like to look at stats of NBA legends to understand why they were so famous and successful. Today, I was reading about Wilt Chamberlain and I saw that in his 1961-1962 season on the Philadelphia Warriors, he had an average of 48.5 minutes per game in a game where there are only 48 minutes per game. I researched the history of the length of NBA games to see if they were longer back then or something, but I found that they've been 48 minutes since the beginning of the NBA. The only reason Wilt averaged more than 48 minutes per game is because of the few games that went into overtime.

On basketball-reference.com, I found that 10 overtime periods were played by the Warriors that season (5 single-overtime games, 1 double-overtime game, and 1 triple-overtime game). Each overtime period is 5 minutes long, so since there were 10 OTs, 50 extra minutes were played.

There were only 80 games in the season at the time, so the number of regulation minutes that season (48X80) was 3,840. With the 50 OT minutes, the total was 3890. Wilt played 3882 minutes that season. 3882/3890 = 99.79% of the total minutes played. Interestingly, I found on Wikipedia that the only reason he didn't play those 8 minutes was that he was ejected from one game with 8 minutes left after getting his second technical foul.

His 3882 minutes over 80 games in a season (3882/80) means he averaged exactly 48.525 minutes per game.

Wilt Chamberlain was 25 years old that season. To compare to a modern NBA superstar, Kevin Durant is 25 years old this season and averages 38.4 minutes per game this season. This number is already ten less than Wilt's average at the same age, and is probably higher than it would have been because Oklahoma City's other superstar Russell Westbrook missed 31 games this season. But even at this pace (in today's 82-game season), Durant would play 3148.8 minutes, over 700 minutes fewer than Chamberlain! This is certainly partially why Chamberlain retired after 14 seasons, while current NBA stars can last much longer. For example, Tim Duncan is in his seventeenth season on the San Antonio Spurs and still plays at an elite level. Because of increasing awareness about injuries and the fact that athletes usually want their career to last as long as possible, I don't think we'll ever see a player play as many minutes as Wilt again.


Works Cited:
http://www.basketball-reference.com/teams/PHW/1962_games.html
http://en.wikipedia.org/wiki/Wilt_Chamberlain
http://espn.go.com/nba/player/_/id/3202/kevin-durant