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17

My answer is probably not very useful when teaching in high school. I'll just mention here a few reasons why this definition is in fact a good one, and why it's a good idea to teach this formula at a university level mathematics course. $\newcommand{\u}{\mathbf{u}}$ $\newcommand{\v}{\mathbf{v}}$ $\newcommand{\w}{\mathbf{w}}$ $\newcommand{\R}{\mathbb{R}}$ Let ...


16

Cosine rule! Think of vectors $\vec a$ and $\vec b$ as two sides of a triangle, with tails at a common vertex. The remaining side is given by $\vec a - \vec b$. Then cosine rule gives us $|\vec a|^2 + |\vec b|^2 - 2|\vec a||\vec b|\text{cos}\,\theta=|\vec a-\vec b|^2$, where $\theta$ is the angle between $\vec a$ and $\vec b$. Now write out $|\vec a|^2$, $|\...


15

It's still not known whether $$\zeta(5) = \sum_{n=1}^\infty \frac{1}{n^5}$$ is a rational number.


14

I think you're correct, strictly speaking, but overreacting. I just checked two calculus books, Rogawski and Strang. Rogawski defines the cross product as the "symbolic" determinant (his words) in question but then defines exactly what he means by this determinant in the same line, so I don't see a problem there. Strang mentions the determinant ...


13

Sure, give up on the mixed-object determinant formula. Instead use tensor arithmetic. Let $\epsilon_{ijk}$ be the completely antisymmetric symbol. Also, while we're at it, let's give up on quaternionic notation for unit-vectors and instead use $\hat{x},\hat{y},\hat{z}$ or $\hat{x_i}$ for $i=1,2,3$ then you have an elegant formula: $$ \vec{A} \times \vec{B} = ...


12

It takes a lot of browsing to find problems somehow related to calculus or analysis, but this is a great MathOverflow list: Not especially famous, long-open problems which anyone can understand. Here are a few from that list: Are there an infinite number of primes $p$ such that the repeating part of the decimal expansion of $1/p$ has length $p-1$? Link. ...


11

You probably get Euler's constant $\gamma$ when you do the integral test comparing $\sum\frac1n$ to $\int\frac{dx}{x}$. Then you can remark that it is unknown whether $\gamma$ is rational.


10

This is a bit obvious I think, but when you introduce sequences and their notation in either an algebra or calculus class, you should certainly show students the Collatz Conjecture as one of the examples.


7

I had a lecturer who thought the same as you, so suggested we just learnt the definition of the cross product (which he derived along the same lines as James S Cook's answer). Nobody liked his lectures because he pitched them way to high for our first-year brains. The point is, it's a very useful mnemonic device, and when you teach it make that clear, but ...


7

this makes no sense But it does. A determinant exists for any $n\times n$ matrix for which the Leibniz formula$$\det A:=\sum_{\sigma\in S_n}\epsilon_\sigma\prod_{i=1}^nA_{i\sigma_i}$$is fully antisymmetric. This is why, for example, quaternionic matrices lack determinants: their elements don't commute. But if we apply it to the case at hand, we don't have ...


7

I strongly dislike the “magnitude and direction” description. Yes, it comes from physics, and is helpful sometimes, but it hides the true nature of vectors, and makes later understanding more difficult (what is the direction of the zero vector?) I prefer to focus from the very beginning on the core property of vectors: they can be added and rescaled; and ...


6

Here's a way to do it by computing the length of $\vec{a} + \vec{b}$ in two different ways: one of which is purely symbolic and the other uses some geometric knowledge. This argument is somewhat similar to Stephan Kubicki's argument, just with a $+$ instead of a $-$ . $\vec{a} + \vec{b}$ can be defined geometrically and component-wise in a straightforward ...


6

My first thought (which is just barely this side of a comment) is that when students are introduced to both vectors and points, they are always with respect to a specific origin. Even if lip service is paid to vectors as a magnitude and direction in some coordinate-free way, realistically the examples have the origin as "the origin". So my conclusion is ...


4

Physically, the problem is to find a third force $\vec F_3$ that balances the two given forces $\vec F_1$ and $\vec F_2$. (This is Newton's Second Law $\vec F_{net}=m\vec a$, where $\vec F_{net}=\vec F_1+\vec F_2+\vec F_3$ and $\vec a=\vec 0$ for equilibrium.) So, $\vec F_3= -(\vec F_1+\vec F_2)$. In Physics classes, this is an opportunity to practice vector ...


4

At the level of standard linear algebra courses, where we consider bases only for finite-dimensional vector spaces, we should work with ordered bases, so that the coefficients of a vector come with an ordering (and thus constitute a vector in $\mathbb R^n$ or $\mathbb C^n$) and so that the matrices that represent linear transformations come with their rows ...


4

I agree with the OP: the determinant is only a mnemonic device, but (a) students may not know how to evaluate a determinant, so it's useless as a mnemonic, and (b) it is not a "legal" determinant and so mathematicallly misleading. I would rather start with $|a \times b| = |a| |b| \sin \theta$ as a contrast to $a \cdot b = |a| |b| \cos \theta$, and ...


2

I'll hazard a somewhat physicsy answer. Consider two vectors $\vec{A}$ and $\vec{B}$. Without loss of generality, choose coordinates where the positive $x$-axis aligns with $\vec{A}$. Hence, $\vec{A} = \langle A, 0 \rangle$. Suppose $\vec{B}$ makes counter-clockwise angle $\theta$ with respect to $\vec{A}$ and denote $\vec{B} = \langle B_1, B_2 \rangle$. See ...


2

The problem is that there isn't a direct connection with area. Sure, you can do a "$\textrm{d}s$" line integral, but that isn't quite the same thing is it? How can we motivate $\int_\gamma P(x,y)\textrm{d}x + Q(x,y)\textrm{d}y$? I think that it is very hard to do. Some physical problems do require this, as you mention, but that seems like a much more ...


2

I remembered GeoGebra doing this several years ago, but I thought that functionality died with the last "classic" version of the program. But, I see it can still be done via the web app. Go to the 3D GeoGebra web app In the input bar, type in some vectors. To type a vector, the notation is: Vector((starting point), (ending point)) You should see a typical ...


2

If you do want this effect (some of the comments are right that there may be other ways to achieve your pedagogical goal), Sage Math has a 3d viewer called "Jmol" which has this ability built in. Try this live example from the documentation, and right-click (Command-click on Mac) to get "Style", then "Stereographic", which will give you a number of options ...


2

Just go with the Cartesian formula. The first time I saw a cross product was in computer code and I always felt like the typical way it's coded is the easiest way to compute $\mathbf{c} = \textbf{a} \times \mathbf{b}$: $c_x = a_y b_z - a_z b_y$ $c_y = a_z b_x - a_x b_z$ $c_z = a_x b_y - a_y b_x$ The pattern is obvious. Each component of $\mathbf{c}$ is the ...


1

It surprises me that nobody has brought up the rule of Sarrus yet: For $\mathbf{u} \times \mathbf{v}$, make the table $\begin{array}{|ccc|cc} \mathbf{i}& \mathbf{j}& \mathbf{k}& \mathbf{i}& \mathbf{j} \\ u_1 & u_2 & u_3 & u_1 & u_2 \\ v_1 & v_2 & v_3 & v_1 & v_2 \end{array}$ , then sum up products of diagonals ...


1

In my opinion $2 \times 2$ matrices, particularly $2 \times 2$ real matrices are fairly easy to understand as mathematical objects, even to folks who haven't been exposed to linear algebra. $2 \times 2$ real matrices can be motivated by linear functions in $\mathbb{R}^2$, (equivalently) as scaled rotations of the plane that fix the origin, as a convenient ...


1

The question of whether formal relationships "make sense" is an old one in mathematics. People asked the question about whether negative numbers "made sense" (how can you start with three pebbles and take away six of them?) or complex numbers (what number when squared gives minus 1?). But we found that putting them in algebraic ...


1

I hope this approach isn't too elementary for your question. I taught myself trig during the summer between my junior and senior year in high school, and developed the following approach. For the "airplane problems" you described, I started with a 4-column table: r | angle | x = r cos(theta) | y = r sin(theta) =================================...


1

The integral of a scalar function on a curve is easily motivated as follows, and doing so is helpful for later motivating the definition of the integral of a vector field along an oriented curve. First, an interval on the real axis can be viewed as a very special sort of curve in the plane. Its length is the integral of the constant function $1$ along this ...


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