22 votes

How do you explain why perpendicular lines have negative reciprocated slopes?

If we have two lines $l$ and $l'$ with slopes $m>0$ and $m'= - \tfrac 1 m$ respectively, then we can always make the following diagram where the blue line is parallel to the $x$-axis and the green ...
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16 votes

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

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 +...
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10 votes

How do you explain why perpendicular lines have negative reciprocated slopes?

One approach is to start with the fact that a 90-degree rotation results from a reflection across the $x$-axis followed by a reflection across the line $y=x$. (In general, reflecting across two ...
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  • 7,575
10 votes

How do you explain why perpendicular lines have negative reciprocated slopes?

One slightly different perspective, but with most of the same mechanics, is to imagine any set of perpendicular lines as having been translated and/or rotated from the x- and y-axes. If you pick any ...
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9 votes
Accepted

How do you explain why perpendicular lines have negative reciprocated slopes?

A proof following from the Pythagorean Theorem: Claim: If two lines are perpendicular, then their slopes are negative reciprocals of one another. Proof: Assume WLOG that two nonvertical lines with ...
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  • 3,765
8 votes

Hands-on demonstration ideas for multivariate calculus

Here's a crafty but perhaps crazy way to convey some ideas to a class. Have all the students gather on the football field (or another field) in a grid on a mildly windy day. Each student carries a ...
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7 votes
Accepted

Equation of a straight line on two dimensional Cartesian plane

Assuming we are talking about an algebra class because of the secondary-education tag, the reason it is okay to use "point-slope form," i.e. $y - y_1 = m(x - x_1)$ as the default way to ...
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  • 19.1k
7 votes

Tips for choosing coordinates of three points such that the coordinates of the orthocenter are integers

An explicit formula for the orthocenter as a function of the three vertices $(x_i,y_i)$, $i=1,2,3$, can be found on Quora, derived by Shambhu Bhat. The orthocenter coordinates are $(o_x,o_y)$, where $...
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6 votes

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

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 ...
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6 votes

How do you explain why perpendicular lines have negative reciprocated slopes?

I find the following approach straightforward geometrically — it's more a demonstration than a formal proof, but it explains the intuition well enough. It's similar to Dag Oskar Madsen's answer ...
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5 votes

Creative problems in 2D vector geometry

I just had some students implement reflection of a ray in a mirrored line segment, which they then used to bounce a light ray around inside a polygon. Here's a crude snapshot:       This project also ...
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5 votes

Creative problems in 2D vector geometry

A modestly non-trivial investigation that yields well to vector algebra is the Euler line. That is, showing that the centroid of any triangle lies on the line segment connecting its circumcenter to ...
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  • 5,539
4 votes

Equation of a straight line on two dimensional Cartesian plane

@TomKern mentioned the "standard" form above. I believe what he has in mind is ax+by=c, which allows for both horizontal and vertical lines. Another form, rarely used, but more like the ...
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  • 17k
3 votes

When are students taught implicit and parametric representations of curves?

I am in a US high school. The implicit equations look like "conics", and are part of the junior (3rd year) class typically called Trigonometry with Algebra. Parametrics are part of the ...
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3 votes

Creative problems in 2D vector geometry

Distance from a line segment to a point? With the right graphing system (maybe Desmos, but it's a bit awkward there) you can use this to draw thick line segments. There's also barycentric coordinates ...
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  • 1,929
3 votes

Creative problems in 2D vector geometry

The proof that the angle between two bodies which collide elastically is $90^o$ is an interesting two dimensional vector algebra problem.
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3 votes

How do you explain why perpendicular lines have negative reciprocated slopes?

I am interpreting the question as: negative reciprocal slopes $\implies$ perpendicular (rather than the converse). (1) If they understand the dot product already, as the projection of one vector on ...
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3 votes

On a special degenerate conic

The general quadric surface in projective space is ruled by two families of lines. These are smooth surfaces. Think about a hyperboloid of one sheet, where is is easy to visualize the two families of ...
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  • 7,575
2 votes
Accepted

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

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} =...
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2 votes

Tips for choosing coordinates of three points such that the coordinates of the orthocenter are integers

The coordinates will be rational numbers, so you just have to pick arbitrary points, then scale by the LCD. This quora question has several answers for what the formula for the orthocenter is; I haven'...
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1 vote

Equation of a straight line on two dimensional Cartesian plane

TL;DR I try not to get overly hung up on notation, and prefer to teach some kind of principle. In this setting, the principle is one of geometric transformation using basic techniques. Many ...
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1 vote

Creative problems in 2D vector geometry

It might be good to prove various trigonometric identities using vector operations. One could use vector methods to derive the laws of reflection and refraction from the Huygens construction. Along ...
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  • 512
1 vote

Creative problems in 2D vector geometry

Here is a problem that I remember running into on an Olympiad which is not too bad with vectors and very hard without. Let $ABCDE$ be a convex pentagon. Let $V$, $W$, $X$, $Y$ and $Z$, respectively, ...
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1 vote

When are students taught implicit and parametric representations of curves?

Neither topic is covered as a Common Core standard as such. Somewhere in middle school, students learn that a circle cannot be a graph of a function because it fails what we call the Vertical Line ...
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  • 5,539
1 vote

When are students taught implicit and parametric representations of curves?

The community college where I teach puts an introduction (lines, circles, ellipses, parabolas) in precalculus, and then covers it again in vector calculus. So, this would be first year and again in ...
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  • 7,686
1 vote

Tips for choosing coordinates of three points such that the coordinates of the orthocenter are integers

I often create problem sets using Excel. I wrote the follow suggestion as a recipe easily developed in Excel. Wherever the digit 1, 2 or 3 follows a letter immediately, the intention is a subscript. ...
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  • 635
1 vote

Hands-on demonstration ideas for multivariate calculus

I like to use ZomeTools to create 3D visualizations of coordinate axes and the interaction between lines and planes. Henry Segerman has produced some amazing looking quadric surfaces.
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