Questions tagged [analytic-geometry]

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1
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3answers
169 views

When are students taught implicit and parametric representations of curves?

Do students learn implicit equations (such as $x^2+y^2-r^2 = 0$) and parametric equations (e.g., $x=a t^2,\;y= 2 a t$) in a first course in algebra, which in the US would be early high school, maybe ...
2
votes
1answer
50 views

On a special degenerate conic

I have a question on MSE that maybe can be better posed here. The question is about degenerate conics, and especially the case of two parallel lines, as in the equation $ 𝑥^2+2𝑥𝑦+𝑦^2=1$. ...
8
votes
5answers
802 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 ...
6
votes
3answers
191 views

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

I want to give my students the coordinates of three points and ask them to find the coordinates of the orthocenter. This is a fairly long problem that can involve finding the equations of the ...
14
votes
7answers
2k views

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

For my purposes, I am interested mostly in a medium-sized liberal art college setting. My students have mostly seen this before, but it is not something they understand. When discussing parallel lines,...
4
votes
0answers
91 views

Making co-ordinate geometry interesting for XI grade students

I am presently teaching eleventh grade (XI standard) students an introductory course in co-ordinate geometry with a focus on preparations for competitive exams. I have seen books like S.L.Loney's co-...
12
votes
3answers
382 views

Hands-on demonstration ideas for multivariate calculus

In teaching Calculus III geometry plays a very important role. It is crucial that students get a good sense of how to visualize curves, surfaces, coordinate axis, frames to curves, vector fields and ...