# Tag Info

### 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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### 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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### 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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### 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 ...

### 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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### 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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### 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 ...

### 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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### 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.

### 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 ...

### 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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Accepted

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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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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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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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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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