Lech Lecha- Vectors and Velocity

"Hashem appeared to Avram and said, 'To your offspring I will give this land.' So he built an altar [in Canaan] to Hashem Who appeared to him. From there he relocated to the mountain east of Beth-el and pitched his tent, with Beth-el on the west and Ai on the east; and he built there an altar to Hashem and invoked Hashem by Name. Then Avram journeyed on, going and traveling toward the south. There was a famine in the land, and Avram descended to Egypt to sojourn there, for the famine was severe in the land." ~Bereishit 12;7-10
"So Avram went up from Egypt, he with his wife and all that was his- and Lot with him- to the south. Now Avram was very heavy with livestock, with silver, and with gold. He proceeded on his journeys from the south to Beth-el, to the place where his tent had been at first, between Beth-el and Ai, to the site of the altar which he had made there at first; and there Avram invoked Hashem by Name." ~Bereishit 13;1-2

Vectors:
In their introductory to Geometry, younger students are introduced to the concepts of points, lines, line segments, and rays.
*A point is a single point or location.
*A line is a straight connection between two points that continues straight and extends indefinitely in both directions
*A line segment is the straight connection between two points with endpoints at the two points
*A ray is the straight connection between two points with an endpoint at one of the points and extending indefinitely through the other point

 A vector is a concept that is not classically introduced to younger students. Essentially, a vector is the same idea as a segment, but it also includes information about distance and tells which direction you're moving in. The two key aspects to a vector are distance (referred to as "magnitude") and direction. Velocity explains the speed at which something is moving (again, it's magnitude) and the direction in which it is moving (23 miles per hour NorthEast). With this understanding of vector and velocity, we can see that velocity is a type of vector measurement, and we can use this understanding to measure how far something travels and the direction in which it is traveling. One simple example would be calculating how far something is traveling either towards (+) or away from (-) a given point. To think about vectors in their simplest form, every straight movement in one direction can be shown as one vector. This means that if I'm traveling in one direction, and then I change my direction, I would need to use 2 different vectors- 1 to explain the first part of my trip, and 1 to explain my trip after I change my direction.

The vector label is based on directions relative to a specific reference point. It's important to realize that this information only tells you have far and in what direction you've moved from your starting point; by itself it doesn't tell you how close or far you are from your reference point. In order to know where you are related to your reference point, you need to add or subtract different vectors to calculate the distance. Using the calculations of multiple vectors, this type of information can be charted on graphs showing distance compared to a location over time on a coordinate plane, and adding the aspect of direction adds an additional quality of information to the graph. For today, we'll just look at simple vectors.

Parsha Connection:
In this week's parsha, we learn of Avram's travels over the course of 25 years. If we look specifically at his travel from the beginning of the 12th chapter through the beginning of the 13th chapter of Bereishit, we see him set-up his tent in Canaan between Beth-el and Ai, travel down to Egypt for the duration of a famine, and then travel back up to return to the exact same location between Beth-el and Ai. How can we use vectors or velocity to explain his trip?

Based on a rough estimate found here, let's assume that Avram's travel distance for his trip from Beth-el down to Egypt was approximately 225 miles. If we designate Beth-el as his starting point (the location where he wants to be) and we use the standard of North and East representing positive directions and South and West representing negative, then his trip down to Egypt is mileage expressed as a negative velocity. So, for example, he starts at 0 miles (when he's standing in Beth-el), and after traveling 100 miles towards Egypt, he had traveled 100 miles SouthWest, or -100 miles, because he was moving away from Beth-el in a negative direction. When he reached his final destination in Egypt, he had traveled 225 miles SouthWest, or -225 miles. On his return trip, he is now heading NorthEast, which is a positive direction. As he traveled back toward Beth-el, let's say he traveled 60 miles per day (velocity = 60 mph NE)- assuming he was traveling by camel.
*After 1 day, he was 60 miles closer to Beth-el, so he had traveled 60 miles NorthEast, or +60 miles
*After 2 days, he was another 60 miles closer, so he had traveled 120 miles NorthEast, or +120 miles
*After 3 days, he was another 60 miles closer, so he had traveled 180 miles NorthEast, or +180 miles
*On his 4th day, he covered the final 45 mi to reach his destination, and he stopped once he reached his 225 miles NorthEast, or +225 miles, to get to Beth-el.

Note that you can see here that by adding the individual vectors from each day, we get an accumulated larger vector, only because Avram was continuing his travel in the exact same direction. 

Extension Thoughts:
If we add the two vectors from each trip together, we get zero - he's back to his initial starting point. (-225 miles) + (+225 miles) = 0 miles. If we add the vectors after the first day of Avram's return trip, his total travels still end up as a negative vector. He travelled 225 miles SouthWest, and then 60 miles NorthEast. (-225 miles) + (+60 miles) = (-165 miles), or 165 miles SouthWest of the vector's starting point.

Everyday Activity:
The concept of vectors can be quite easily integrated into younger students learning about measurement and distance. Simple classroom activities can include students measuring distance from a reference point and then having them walk certain distances in directions related to your reference point and expressing their walks using vector language.

A simple everyday example of vectors: Have you ever paid attention to the highway distance markers as you drive along? Using those markers, you can say that you've driven 23 miles North toward your destination.

Toldot- A Timeline Through Deductive Reasoning

Deductive reasoning: a logical process in which a conclusion drawn from a set of premises contains no more information than the premises taken collectively (ref. dictionary.com)

When trying to problem solve, deductive reasoning helps us logically move from one step to the next. Deductive reasoning allows us to look at the information that we are provided with and state, with certainty, that the next logical step can also be relied on, because the original information proves the next step. Each step that is proven through deductive reasoning can then be relied on as a fact to support our logical thinking towards a final solution to the problem at hand.

As we've been reading through the parsha each week, we have followed storylines which offer some concrete sense of time. However, taken together, how much time is actually passing between each phase of each story? If we take the individual pieces of information that we are provided with from the text and Rashi's commentaries, we can use deductive reasoning to calculate some of the missing pieces of information in order to put together a fuller picture for ourselves. Is the story between the parshiot a straight timeline, or is it broken into parts dealing with each "main character"? With some deduction, we can find out. (My completed storyline for the primary Toldot "characters" with some added information appears at the bottom of this post. Check your calculations and timeline against mine and see how they match-up. Can you fill in the extra information that I calculated?)

Whenever sorting information for problem solving, the first step is to map out the information that you have.

• From Lech Lecha 17;17 and Vayeira 21;2 we know that when Isaac was born, Sarah was 90 years old and Abraham was 100 years old
• From Chayei Sarah 23;1 we know that Sarah lived to be 127 years old
• From Rashi's commentary on Sarah's death, we know that she died upon hearing the news that Isaac was to be sacrificed at the akedah
• From Rashi on Toldot 25;20, we know that Rebecca was born in the year of the akedah
• From Chayei Sarah 25;7, we know that Abraham lived to be 175 years old
• From Toldot 25;20, we know that Isaac was 40 years old when he married Rebecca
• From Toldot 25;26, we know that Isaac was 60 years old when Esau and Jacob were born
• From Rashi's commentary on Toldot 25;27, we know that Jacob bought Esau's birthright when they were 13 years old
• From Rashi's commentary on Toldot 27;2, we know that Jacob took Esau's blessing from Isaac when Isaac was 123 years old
• From Toldot 26;34, we know that Esau was 40 years old when he married Judith and Basemath

The second step to problem solving is to identify the questions that you are trying to answer.
--How old was Isaac at the akedah (and Sarah's death)?
--How old was Isaac at Abraham's death?
--How old was Rebecca when she married Isaac?
--How old was Rebecca when Esau and Jacob were born?
--How old were Isaac and Rebecca when Esau sold his birthright?
--How old were Jacob, Esau, and Rebecca when Isaac gave Jacob Esau's blessing?
--How old were Rebecca and Isaac when Esau was married?
--How old is Jacob (and Esau) at the end of Toldot?

Now let's see what information we can use to answer these questions:
--How old was Isaac at the akedah (and Sarah's death)?

• Sarah was 90 when Isaac was born, and she was 127 when she heard of the akedah. 
• 127-90=37, so Isaac was 37 at the akedah and when Sarah died

--How old was Isaac at Abraham's death?

• Abraham was 100 when Isaac was born, and Abraham lived to be 175.
• 175-100=75, so Isaac was 75 when Abraham died

--How old was Rebecca when she married Isaac?

• Isaac was 37 at the akedah, which was the same year that Rebecca was born.
• Isaac was 40 when he married Rebecca.
• 40-37=3, so Rebecca was 3 when they were married (keeping strictly to math here, no societal norms commentaries)

--How old was Rebecca when Esau and Jacob were born?

• Isaac was 60 when they were born
• Isaac was 37 years older than Rebecca
• 60-37=23, so Rebecca was 23 when they were born

--How old were Isaac and Rebecca when Esau sold his birthright?

• Isaac was 60 when they were born; Rebecca was 23 when they were born
• Esau and Jacob were 13 when the birthright was sold
• 60+13=73, so Isaac was 73; 23+13=36, so Rebecca was 36

--How old were Jacob, Esau, and Rebecca when Isaac gave Jacob Esau's blessing?

• Isaac was 60 when they were born; Rebecca was 23 when they were born
• Isaac was 123 when he gave Jacob Esau's blessing
• 123-60=63, so Jacob and Esau were 63; 23+63=86, so Rebecca was 86

--How old were Rebecca and Isaac when Esau was married?

• Isaac was 60 when they were born; Rebecca was 23 when they were born
• Esau was 40 when he was married
• 60+40=100, so Isaac was 100; 23+40=63, so Rebecca was 63

(click on the picture for a larger view of the timeline)

From this timeline, we can see that, although most of the storylines in these few parshiot are in order, there are some pieces of information that are placed out of order, seemingly to keep them together with the sections that discuss the particular "character" in more depth.