Colinear vs Parallel
Nathan Kamgang
What does it means to be going in the same direction. If we were both walking in the same room with you, I’m sure you could determine intuively wheter we’re going in the same direction. How can we catch that intuition precisely. We give coordinate to our space using the cartesian plane. What’s the difference between these two sets of vectors:
If you imagine the unique line on which a vectors lies, We are on the same direction if we are on the same line [[]]
Our simplistic model doesn’t hold direciton is more than that. If I we place our selves on both sides of a rails track. We aren no t on parallel line, bu t were are follwing diffente irhections. YOu’re not obliged to draw vectors at eh origin, you can draw them anywhere on the graph. Any vectors not at the origin can be translated back Even when weren’t colinear(on the same line), we can still go in the same direction as long as we’re parallel.
[Two parallel line and two non prallel lines]
Conclusion
Parallel vectors go in the same direction. If the 2 vectors are jon the same line, they are said to be colinear.
How to know if two vectors are paralle(Place a link to the article about position vs direction vectors)
- You know how to add vectors visually using the triangle method
position vs direction vector
Every vector has a position vector even vector at the origin. we omit for vector a the origin because it easier
Visually, a vector is a line segment with a tip and a tail in the 2D plane. The vector is a straight line, so if I give you where the tip of the coordinate of the tip of the vector and the coordinate of the tail, you only have to draw a straight line between those two points to draw our vector or line segment.
(Picture showing two coordinate turning into a vector)
If the vector is at the origin. The tip is at (0,0). There is no need to specify that each time, thus we only need the coodinate of the tip.
(Picture drawing a vector at the origin)
Physically, we can move in the same direction by walking in parallel line. or by walking in the same line and I can
If two racing cars A and B races are moving east during a race in a straight line. Intuitevely, we say that the car are moving in the same direction. Consider a military march where soldier are moving in the same line forward. We consider them to be walking in the same direction too. Whereas the soldier are walking on the same line, and the car speeds on parallel and different lines, we say that they are both going in the same direction. How to define direction so that it encompasses, those two scenarios?
The answer is the direction vector. It’s a vector centered at the origin of the plane that specify the direction in which vector are going. If a vector is vector is at the origin. We consider it to also be its own direction vector. What we should we though when the vector is not. Remember from our intuition, we want the both two parallel vector to have the same direction. As such, we must define the direction vector in a way that makes two parallels vector codirectional. (Picture showing two parallel vector)
In order to find the direction. Just take that position vector and translate it toward the origin. You keep the size and orientation of the vector the same. The translated vector at the origin is the direction vector. (Picture showing this translation process). Now if two vector have the same orientation but aren’t colinear like our two cars earlier, their direction vector will point in the same direction.
Algebraically if u and v and position vectors, the direction vector is:
Convice yourself using vector addition by triangle or parallelogram method that u-v is indeed the direction vector we described
Explain briefly why we use the unit direction vector to then speicify the direction precisely.
If you want to graphs vectors visually you must understand the nuance between a direction vector and a position vector. To draw a 2 dimensional position vector, draw a dot anywhere in the 2D plane Insert picture 1 []]
. A direction vector always start at the origin of the plane.
If you don’t want to draw you can specify a direction vector using a coordinate like (2,3) which represent the same point we draw earlier.
Now how will you represent this vector Insert picture 2 [[static/images/2.pdf|2]]
Any vector has an origin which is the point where you begin the draw the vector and a terminal point(where you stop drawing the vector). The origin of any position vector is at the origin of the plane. So if the vector isn’t at the origin only one position vector won’t cut. The solution? Use 2 directions vectors to draw a vector not at the origin.
One position vectors equals the origin of the vector not at the origin and the other position vector indicates the terminal point of the same vector. Insert picture three
What if you need the lenght of your position vector?" Add image4 [[images/4.pdf|4]]
A vector not at the origin has two position vectors instead of one so the pythagora can’t be used in the same way to find the lenght of the position vector. We need a new tool the direction vector.
Substract one direction vector from the other and you get the direction vector. Algebraically, you can substract their corresponding coordinate. Insert the definition of direction vector Visually it gives:
Insert 5
We can move a direction vector around the plane as long as we don’t change its lenght of direction. That’s why a direction vector is also called a free vector. IN the next picture we translate a direction vector so that it falls between u and w. Look at the next picture
Note that the translated vector has the same lenght and direction as the original orange vector.
Hey you! yes i’m talkking to you take a time to look at the above graph again because that’s important. If w and u are our position vector, to get the lenght or direction of the vector not at the origin, we can use the lenght and direction of the free direction vector. Because that direction vector is parallel to our vector not at the origin.
There you have it. Position vector and Vector not centered at the origin and direction vectors
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