Questions tagged [analytic-geometry]

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2 votes
3 answers

Equation of a straight line on two dimensional Cartesian plane

What is the most appropriate way to write equation of a straight line through a given point $(\alpha,\beta)$? If you write it using the standard form $y- \beta = m (x - \alpha )$ Where m is the ...
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7 votes
7 answers

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 ...
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1 vote
3 answers

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 ...
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2 votes
1 answer

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$. ...
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8 votes
5 answers

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 ...
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  • 602
6 votes
3 answers

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 ...
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  • 635
14 votes
7 answers

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,...
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4 votes
0 answers

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-...
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  • 141
12 votes
3 answers

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 ...
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