Vectors: Suppose 2D line passes through P0(10, 15) and P1(200, 20)...

[HugeLag]

Suppose a 2D line passes through two points P0(10, 15) and P1(200, 20). Answer Questions 7 *- 10 regarding this line.

Question 7)

Which of the following vectors parallel to the line?

(180, 10)

(190, 5)

(380, 10)

(180, 10)

Can someone please give me an explanation on how to work things like this out? I am not asking for straight answers, I actually want to learn this, but I cant seem to really find stuff online that relates to this unless I am searching for the wrong things? I know its about placement lines. I am seriously confused, am I meant to draw a 200x200cm grid, plot the answers given and see which ones are parallel to the two points P0(10, 15) and P1(200, 20)?

All explanations and help is much appreciated!!!

 

[Deleted member 4993]

Suppose a 2D line passes through two points P0(10, 15) and P1(200, 20). Answer Questions 7 *- 10 regarding this line.

Question 7)

Which of the following vectors parallel to the line?

(180, *10)

(190, 5)

(*380, *10)

(*180, 10)

Can someone please give me an explanation on how to work things like this out? I am not asking for straight answers, I actually want to learn this, but I cant seem to really find stuff online that relates to this unless I am searching for the wrong things? I know its about placement lines. I am seriously confused, am I meant to draw a 200x200cm grid, plot the answers given and see which ones are parallel to the two points P0(10, 15) and P1(200, 20)?

All explanations and help is much appreciated!!!

What are the slopes of each of those vectors?

What is the slope of the given line?

 

[pka]

Suppose a 2D line passes through two points P0(10, 15) and P1(200, 20). Answer Questions 7 *- 10 regarding this line.
Question 7)
Which of the following vectors parallel to the line? (180, *10) STRANGE!

In many years of teaching courses involving vectors, I have never seen this strange notation.
If the two points are standard ordered pairs the that line has a direction vector of \(\displaystyle <190,5>\).
Any line with \(\displaystyle <190,5>\) as a direction vector is parallel to the given line.

I have absolutely no idea what \(\displaystyle (180,~^*10)\) could mean!

 

[HugeLag]

In many years of teaching courses involving vectors, I have never seen this strange notation.
If the two points are standard ordered pairs the that line has a direction vector of \(\displaystyle <190,5>\).
Any line with \(\displaystyle <190,5>\) as a direction vector is parallel to the given line.

I have absolutely no idea what \(\displaystyle (180,~^*10)\) could mean!

TYPO! ahahah sorry

there is no *.

 

[HugeLag]

Sorry the *'s were a typo my bad. I edited the original post.

 

[pka]

Sorry the *'s were a typo my bad. I edited the original post.

Well then if

Suppose a 2D line passes through two points P0(10, 15) and P1(200, 20).

\(\displaystyle P_0: (10, 15)~\&~ P_1: (200, 20)\) is standard point notation then I gave you the correct answer in reply #3.

 

[HugeLag]

Well then if

\(\displaystyle P_0: (10, 15)~\&~ P_1: (200, 20)\) is standard point notation then I gave you the correct answer in reply #3.

Thanks1

one more question!

Which of the following equations represent the line?

5x - 190y +2800=0

5x + 6y -8=0

(x-10, y-15)•(-5, 190)=0

(x-200, y-20)•(5, -190)=0

I worked this out using m=y2-y2/x2-x1

My answer was

y=1over38x+14 14over19

This seems wrong, because it does not match with the answers given in the question? This is in relation to the points P0 (10,15), and P1(200,20)

 

[Ishuda]

Suppose a 2D line passes through two points P0(10, 15) and P1(200, 20). Answer Questions 7 *- 10 regarding this line.

Question 7)

Which of the following vectors parallel to the line?

(180, 10)

(190, 5)

(380, 10)

(180, 10)

Can someone please give me an explanation on how to work things like this out? I am not asking for straight answers, I actually want to learn this, but I cant seem to really find stuff online that relates to this unless I am searching for the wrong things? I know its about placement lines. I am seriously confused, am I meant to draw a 200x200cm grid, plot the answers given and see which ones are parallel to the two points P0(10, 15) and P1(200, 20)?

All explanations and help is much appreciated!!!

It sometimes helps to see where statements are coming from, i.e. vectors with the same (scaled) direction vector are parallel. Where this comes from is lines with the same slope are parallel.

If we look at two points such as (0,1) and (1,3), we can write the point/slope form of the line passing through the points as
y = \(\displaystyle \dfrac{3-1}{1-0} (x-0) + 1 = 2 x + 1.\)
Thus any line with a slope of 2 is parallel to that line.

Now suppose we have the ordered pair of vectors <0,1> and <1,3> forming a line. The direction vector of that pair of ordered vectors is <\(\displaystyle 1-0,\, 3-1\)>=<1, 2>. If we had another ordered pair of vectors with a direction vector vector <1 * a, 2 * a> for some non-zero a, we see that the slope for the line between those two vectors would be \(\displaystyle \dfrac{2a}{1a}\) = 2. Thus the [continued] line between those two vectors must be parallel.

One final thing. When one speaks of a direction vector such as I did above, I was using short hand for 'the direction vector from <0,1> to <1,3> is ...', that is a specific direction vector is always from one point to a different point. By convention, if the first point is not mentioned, that point is the origin. So when one says the direction vector for the vector <180, 10> is <180, 10> they really mean the direction vector from <0, 0> to <180, 10> is <180, 10>.

 

[pka]

It sometimes helps to see where statements are coming from, i.e. vectors with the same (scaled) direction vector are parallel. Where this comes from is lines with the same slope are parallel.
If we look at two points such as (0,1) and (1,3), we can write the point/slope form of the line passing through the points as
y = \(\displaystyle \dfrac{3-1}{1-0} (x-0) + 1 = 2 x + 1.\)
Thus any line with a slope of 2 is parallel to that line.
Now suppose we have the ordered pair of vectors <0,1> and <1,3> forming a line. The direction vector of that pair of ordered vectors is <\(\displaystyle 1-0,\, 3-1\)>=<1, 2>. If we had another ordered pair of vectors with a direction vector vector <1 * a, 2 * a> for some non-zero a, we see that the slope for the line between those two vectors would be \(\displaystyle \dfrac{2a}{1a}\) = 2. Thus the [continued] line between those two vectors must be parallel.

May I ask what any of the above has to do with the original posting? It is perfectly clear to those of us involved with the promotion of vector geometry, this exercise is specifically designed to avoid the use of slope. Slope in \(\displaystyle \mathbb{R}^3\) has no meaning, but direction vectors do. Thus the idea is to give students a unified concept of linear relations.

 

[Ishuda]

May I ask what any of the above has to do with the original posting? It is perfectly clear to those of us involved with the promotion of vector geometry, this exercise is specifically designed to avoid the use of slope. Slope in \(\displaystyle \mathbb{R}^3\) has no meaning, but direction vectors do. Thus the idea is to give students a unified concept of linear relations.

Now pka, I don't think anything is ever perfect The idea behind my post was to introduce the idea of a direction vector [in 2-D] with something a student is already familiar with. As far as slope having no meaning in \(\displaystyle \mathbb{R}^3\) [or in \(\displaystyle \mathbb{R}^n\)], I disagree. What is the partial derivative of an n-linear map but the slope of the line when all variables but one is held constant. The parts of the n dimensional directional vector can be related to the n dimension 'vector gradient of an n-linear map'.

It may be that for some, a better understanding about some one thing can be obtained by relating it to some other thing already known. With the "All explanations and help is much appreciated!!!" statement, I thought that this might be the case here. Of course, I might have been incorrect but I still don't think it was wrong to try [especially in view of the follow up question "Which of the following equations represent the line?"].