Tag Archives: math

Angle between two lines in the plane……Vector product in 3D…….connections???????

So I was in the middle of converting my geometry application Geostruct (we used to call them programs) into javascript

 get it here with the introduction .doc file here

when I decided that the “angle between two lines” routine needed a rewrite. Some surprises ensued !

angle between two lines pic1

Two lines,  ax + by + c  = 0  and  px + qy + r = 0

Their slopes (gradients) are  -a/b = tan(θ)  and  -p/q = tan(φ)

The angle between the lines is  φ – θ,

so it would be nice to know something about  tan(φ – θ)

Back to basics, where  tan(φ – θ) = sin(φ– θ)/cos(φ– θ),

and we have the two expansions

sin(φ– θ) = sin(φ)cos(θ) – cos(φ)sin(θ)   and

cos(φ– θ) = cos(φ)cos(θ) + sin(φ)sin(θ)

So we have  tan(φ – θ) = (sin(φ)cos(θ) – cos(φ)sin(θ))/( cos(φ)cos(θ) + sin(φ)sin(θ))

Dividing top and bottom by  cos(φ)sin(θ)  and skipping some tedious algebra we get

tan(φ – θ)   =  (tan(φ) – tan(θ))/(1 +  tan(φ)tan(θ))

This is where the books stop, which turns out to be a real shame !

Going back to the two lines and their equations, the two lines

ax + by = 0  and  px + qy = 0

have the same angle between them (some things are toooo obvious)

Things are simpler if we look at these two lines through the origin when they both have positive slope.

Take b and q as positive and write the equations as   ax – by = 0  and  px – qy = 0

Then the point whose coordinates are (b,a) lies on the first and (q,p) lies on the second.

angle between two lines pic2

Also, the slopes of the two lines are now  a/b , tan(θ)   and  p/q , tan(φ)

Let us put these into the  tan(φ – θ)   equation above, and once more after tedious algebra

tan(φ – θ)  = (bp – aq)/(ap + bq)

which is a very nice formula for the tan of the angle between two lines.

This is ok if we are interested just in “the angle between the lines”,  but if we are considering rotations, and one of the lines is the “first” one, then the tangent is inadequate. We need both the sine and the cosine of the angle to establish size AND direction (clockwise or anticlockwise).

The formula above can be seen as showing  cos(φ– θ)  as  (ap + bq) divided by something

and  sin(φ– θ)  as  (bp – aq) divided by the same something.

Calling the something  M  it is fairly clear that    (ap + bq)2 + (bp – aq) 2 = M2

and more tedious algebra and some “observation and making use of structure” gives

M= (a2 + b2)(p2 + q2)

and we now have

sin(φ– θ)  = (bp – aq)/M  and  cos(φ– θ) = (bq + ap)/M

and M is the product of the lengths of the two line segments, from the origin to (b,a) and from the origin to (q,p)

It was at this point that I saw M times the sine of the “angle between” as twice the well known formula for the area of a triangle. “half a b sin(C)”, or, if you prefer, the area of the parallelogram defined by the two line segments.

Suddenly I saw all this in 2D vector terms, with bq + ap being the dot product of (b,a) and (q,p) , and bp – aq as being part of the definition of the 3D vector or cross product, in fact the only non zero component (and in the z direction), since in 3D terms our two vectors lie in the xy plane.

Why is the “vector product” not considered in the 2D case ??? It is simpler, and looking at the formula for sine , above, we have a 2D interpretation of the “vector”or cross product as twice the area of the triangle formed by (b,a) and (q,p). (just as in the standard 3D definition, but treated as a scalar).

So “bang goes” the common terms, scalar product for c . d  and vector product for  c X d

Dot product and cross product are much better anyway, and a bit of ingenuity will lead you to the reason for the word “cross”.

This is one of the things implemented using this approach:

gif rolling circle

Anyway, the end result of all this, for rotating points on a circle, was a calculation process which did not require the actual calculation of any angle. No arctan( ) !

Leave a comment

Filed under geometry, language in math, teaching

School——>PARCC tweet———>suspension———->and then?????

Thank you Audrey Watters for leading me to this exposure of the behaviour of testing corporations.

These two are MUST READs, and should be passed on to everybody:

http://www.bobbraunsledger.com/breaking-pearson-nj-spying-on-social-media-of-students-taking-parcc-tests/

http://kengreenwood.com/pearson_spying_on_kids/

Leave a comment

Filed under big brother, education, testing

Language in math, again.

“Is” is a very overworked word, to the point of illogicality.

which of these both

Technically in both cases none of them.

In everyday language we can get away with the question and accept the answer “The first one” even though it is actually a picture of the head of a dog.

In math we MUST be more precise, and ask “Which of these graphs is the graph of a function?”, or “Which of these graphs could represent a function?”.

A graph is NEVER a function, and a function is not a graph. If we actually followed the Common Core on this it would be even more troublesome, as a graph is DEFINED as a set of ordered pairs as below —
…………………………………………
Functions 8.F
Define, evaluate, and compare functions.
1. Understand that a function is a rule that assigns to each input exactly
one output. The graph of a function is the set of ordered pairs
consisting of an input and the corresponding output.
…………………………………………
But at least WE all know what a graph is…..or do we?

4 Comments

Filed under language in math, teaching

Geometry problem, borrowed and extended. Try it !

I borrowed this from http://fivetriangles.blogspot.com/
184. *** Overlapping sectors
overlappingsectorsspic
In the diagram is rectangle ABCD with height 10 cm. An arc with centre at point B is drawn from point A to side BC. An arc with centre at point C is drawn from point D to side BC. Given that the shaded (coloured) regions ⓐ and ⓑ have equal area, determine the length of BC.

The extension is “and where do the quarter circles cross?”
It could be a two line calculation !

Leave a comment

Filed under geometry, teaching

Calculus without limits 3

So you tried, or you didn’t, now here is the derivative of 1/sqr(x)

calculus3

Leave a comment

Filed under algebra, calculus, education, teaching

Limits

George, to his teacher:

I have now integrated my preconceived ideas and the enlightenments engendered by yourself, but I still have trouble differentiating between “the limit of” and “the limits of”.

George’s teacher, aside:

I think George would be better off writing a novel. he could call it “The Limits of Continuity”.

Leave a comment

Filed under calculus, education, humor, language in math, teaching

“I did my best to pass the test”

I did the sums, no hesitation.
But then it asked for explanation.
“I know it’s right”, I wrote down fast,
“I understood from first to last!”.
“I’m going to be a mathematician,
“Not a fingernail technician!”.

Leave a comment

Filed under algebra, arithmetic, calculus, education, fractions, geometry, humor, verse

Calculus without tears (that is, without limits)

“As h approached zero I reached the limit of my understanding.”

So it seemed to me that calculus without limits would be a good idea.

Not just for powers of x, but also for trig, exp and log functions.

This is the first of several posts on this subject.

calculus1 pic

calculus1

Leave a comment

Filed under algebra, calculus, geometry, teaching

Infinity, a place beyond.

That most strange place, infinity,
Is somewhere I don’t want to be.
I’d rather stay with Brouwer
In his ivory tower.

 

and for something lighter try Heavy Man

 

Leave a comment

Filed under abstract, arithmetic, geometry, humor, language in math, verse

Lament to the Common Core geometry

Could I move this trapezoid
To that one, in the endless void?
I tried translation and rotation.
Then I had a crazy notion.
I would pass a rigid motion.
Result – a lovely hemorrhoid.

Leave a comment

Filed under geometry, humor, language in math, verse