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Questions tagged [vectors]

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25 votes
11 answers
6k views

Why do we still teach the determinant formula for cross product? And is it as bad as I think it is?

The cross product is an important vector operation in in any serious multivariable calculus course. In most textbooks that I'm aware of, right after the definition, we always introduce the ...
user avatar
17 votes
4 answers
3k views

What are some of the open problems that can be suitably introduced in a calculus course?

I feel it may be a good idea to introduce some related open problems in a calculus course. Surely I am not expecting my students to solve any one of them, though I cannot say it is absolutely ...
Zuriel's user avatar
  • 4,275
10 votes
2 answers
338 views

3D vectors with 3D glasses?

I am going to teach 3D vectors soon to my A-level students. Last time I taught it a lot of them struggled to visualise the angle between two vectors in 3D space. I had an idea that I could get them ...
harpomiel's user avatar
  • 201
7 votes
5 answers
878 views

How do you plausibly explain that the geometric and the coordinate expressions for the scalar product are equivalent?

The standard scalar product on $\mathbb{R}^3$ is defined via $$\vec a\cdot\vec b := a_1b_1+a_2b_2+a_3b_3$$ On the other hand, it can be expressed in a more geometrical way through the lengths of the ...
Photon's user avatar
  • 602
7 votes
7 answers
520 views

Creative problems in 2D vector geometry

What are some "interesting" and creative problems or exercises on specifically 2-dimensional vector geometry that a high school student might find compelling to solve? The class' current ...
Damian Reding's user avatar
7 votes
3 answers
275 views

Why is it difficult to freely change between points and vectors?

I have noticed working with bright undergraduates that it is not uncommon for them to have difficulty easily converting between a point—say, a point $p$ on a surface $S \subset \mathbb{R}^3$&...
Joseph O'Rourke's user avatar
6 votes
1 answer
170 views

Difficulty in teaching the coordinates of a vector with respect to a basis $\{v_1,v_2,\ldots,v_n\}$

Let $V$ be a finite dimensional vector space and let $B=\{v_1,v_2,\cdots,v_n\}$ be a basis of $V$. If a vector $v$ can be written as $$v=a_1v_1+a_2v_2+\cdots+a_nv_n,$$ we call $(a_1,a_2,\cdots,a_n)$...
Zuriel's user avatar
  • 4,275
5 votes
2 answers
281 views

Presenting ways to find a resultant force

To begin with I am working in a high school classroom where the students are working on the applications of vectors. The beginning of the lesson is about calculating direction and magnitude of vectors ...
Alex Rost's user avatar
3 votes
4 answers
462 views

Why use the vague notion of "vector" when you have $\mathbb R^2,\mathbb R^3,\mathbb R^4,\ldots$?

I'm reading an introductory course on groups. In this course, the author illustrates concepts using the vectors of the plane. For example, "the set of vectors in the plane(or in space) is a group ...
Stéphane Jaouen's user avatar
2 votes
4 answers
500 views

Teaching Clifford Algebra Instead of Imaginary/Complex Numbers

For those unaware, Clifford Algebra (also known as Geometric Algebra) is able to generalize vectors and rotations in n-dimensional space, and simplifies a great many formulas. However, I was curious ...
johnnyb's user avatar
  • 1,249
2 votes
4 answers
403 views

How can one lone picture prove the Triangle Inequality, $|x−y|≤|x|+|y|$, $|x|−|y|≤|x−y|$, and the Reverse Triangle Inequality?

I always showcase separate pictures of Triangle Inequality, and Reverse, to 16-years-old students in 1st class. I reshow pictures in 2nd class. I preachify Please remember these 4 inequalities. ...
user27289's user avatar
  • 139
2 votes
2 answers
455 views

Line Integral Motivation

Is there a case to be made that the topic of line integrals should only involve vector fields? My colleagues and our textbook take the position that line integrals should only be taught from a vector ...
Nick C's user avatar
  • 9,639
2 votes
0 answers
98 views

Teaching linear algebra, wacom tablet display of coordinate system, eigenvectors, markov chains

I am teaching linear algebra as part of an information retrieval course, which now occurs online. I have a Wacom tablet and free drawing software, sketchBook for artists, so can draw circles ellipses ...
ExpressionCoder's user avatar
0 votes
2 answers
162 views

Multiple proofs for the same problem

One way of encouraging students to explore mathematics can be letting them to use different approaches to solve the same problem. If students can find alternatives from different areas of mathematics ...
Janaka Rodrigo's user avatar