Collinear Points

[harpazo]

Using the result obtained in the thread Equation of A Line, show that the three distinct points (x_1, y_1), (x_2, y_2), and (x_3, y_3) are collinear (lie on the same line) if and only if the determinant shown in the picture equals 0.



1. Seeking the first-two steps and hints.

2. Why does Sullivan ask such a question in course that is not linear algebra?

3. What relationship does geometry have with the determinant?

 

[Deleted member 4993]

Using the result obtained in the thread Equation of A Line, show that the three distinct points (x_1, y_1), (x_2, y_2), and (x_3, y_3) are collinear (lie on the same line) if and only if the determinant shown in the picture equals 0.

View attachment 13854

1. Seeking the first-two steps and hints.

2. Why does Sullivan ask such a question in course that is not linear algebra?

3. What relationship does geometry have with the determinant?

1. Seeking the first-two steps and hints...........................................................replace the values of x₁, x₂. etc. with given values - and calculate the determinant.

2. Why does Sullivan ask such a question in course that is not linear algebra? Ask Mr. Sullivan - I cannot read his mind. I do not feel that these questions are related to linear algebra. These are just "arithmetic" manipulations.

3. What relationship does geometry have with the determinant? - What relationship does "area of a triangle" have with the process of "multiplication".

 

[Otis]

… Why does Sullivan ask such a question in course that is not linear algebra?

I don't think of math texts as courses, per se, but I can tell you that a single textbook can service more than one course. It could be that Sullivan included extra material to accommodate instructors who are teaching an algebra course designed for students who plan to enroll in a linear-algebra class. That way, some basic concepts can be introduced, to give such students an advantage. Instructors teaching other algebra courses may very well skip some extra chapters in the book.

I remember paying over a hundred bucks for a math textbook in one of my courses; I'm sure that book contained at least a dozen chapters. We used three of them, in class.

Here's something you can try. Look over the material in Sullivan's book that appears before chapter one. He may have included some rationale behind the purpose of his book or its organization. That is, maybe he has answered your question above, in one of those pre-sections.

?

 

[harpazo]

1. Seeking the first-two steps and hints...........................................................replace the values of x₁, x₂. etc. with given values - and calculate the determinant.

2. Why does Sullivan ask such a question in course that is not linear algebra? Ask Mr. Sullivan - I cannot read his mind. I do not feel that these questions are related to linear algebra. These are just "arithmetic" manipulations.

3. What relationship does geometry have with the determinant? - What relationship does "area of a triangle" have with the process of "multiplication".

Wish I could contact Sullivan, trust me.

 

[harpazo]

I don't think of math texts as courses, per se, but I can tell you that a single textbook can service more than one course. It could be that Sullivan included extra material to accommodate instructors who are teaching an algebra course designed for students who plan to enroll in a linear-algebra class. That way, some basic concepts can be introduced, to give such students an advantage. Instructors teaching other algebra courses may very well skip some extra chapters in the book.

I remember paying over a hundred bucks for a math textbook in one of my course; I'm sure that book contained at least a dozen chapters. We used three of them, in class.

Here's something you can try. Look over the material in Sullivan's book that appears before chapter one. He may have included some rationale behind the purpose of his book or its organization. That is, maybe he has answered your question above, in one of those pre-sections.

?

Good idea. Nice reply. Thank you.

 

[harpazo]

Using the result obtained in the thread Equation of A Line, show that the three distinct points (x_1, y_1), (x_2, y_2), and (x_3, y_3) are collinear (lie on the same line) if and only if the determinant shown in the picture equals 0.

View attachment 13854

1. Seeking the first-two steps and hints.

2. Why does Sullivan ask such a question in course that is not linear algebra?

3. What relationship does geometry have with the determinant?

What given values do I replace x_1 and y_1, x_2, y_2, x_3, and y_3 with?

 

[Deleted member 4993]

What given values do I replace x_1 and y_1, x_2, y_2, x_3, and y_3 with?

I was "hoping" you would realize that this is NOT a plug-and-chug problem!

When the vertices of a triangle are collinear - what is the height of the triangle?

What would be the area of that triangle?