The two right-angled triangle can seen. We will look at which has a side on x axis. We can clearly see from Pythagorean theorem that sin²(Ɵ)+cos²(Ɵ)=1.
But what manipulation of space. Well , to do that we need to see points as two separate ( cause of two dimension) real numbers. Like (x,y). Now we can do basic transformations along axes. Like if we want to move along x-axis 1 unit just add 1 to a and new point is (a+1,b) . You can do same thin c unit along y-axis by just adding c to b and you will get point (a,b+c). This is boring I know. Then what about rotating a point ?
Sounds cool I guess! But to rotate a point in space we first need to change our view a bit. Think about point (1,0) , if we rotate them Ɵ degree. The new point will be (cos(Ɵ),sin(Ɵ)) and then think about (0,1) this is already 90 degree rotated one of (1,0) when you think about. Then it will be just (cos(90+Ɵ),sin(90+Ɵ)) = (-sin(Ɵ),cos(Ɵ)). Now we had rotated most simple points except (0,0) which won’t change. Now we can do same process to the point (2,0) after the Ɵ degree rotation we will get (2cos(Ɵ),2sin(Ɵ)) and for the point (0,3) after the Ɵ degree rotation we will get (-3sin(Ɵ),cos(Ɵ)).
But how could we rotate a point which has two non-zero real number like (2,3) for Ɵ degree around to origin? Well this has some trick that we know already. Think about x and y axis is rotated some angle Ɵ.
We know how to calculate a’ and b’ .
a’ = (a.cos(Ɵ),a.sin(Ɵ)) and b’ = (-b.sin(Ɵ), b.cos(Ɵ)). Like we discus before. If we add these x and y values we get P’. You could say how but it is just like what we do before. When we need to present point (3,4) we would go 3 point on x axis and 4 point on y axis. Pretty straightforward right. Here what we do go along x’ to the a’ and along y’ to the b’. This is the same process adding them together. P’ = a’+b’ = (a.cos(Ɵ)-b.sin(Ɵ),a.sin(Ɵ)+b.cos(Ɵ) ). I actually do vector addition here. Thinking that a’ and b’ is a 2-d vector is almost same as think them as a point in 2d space.
Now you now how to rotate a point in counter clock wise around origin. Just using -Ɵ you can rotate clock wise too. Vectors are objects whose have direction and magnitude. Magnitude is simply length of a vector. And calculating this length is same as calculating distance between point and origin. For example (3,4) is sqrt(3²+4²) unit away from origin.
Let v=(x,y) is vector and a is real number. Then v.a =(a.x,a.y). let u is another vector and b is another real number and u =(z,t) then
v.a+u.b=(a.x+b.z,a.y+b.t) .This new vector is linear combination of u and v.
Another important thing is with combination of two vector with different directions and non-zero length, you can get every vector in two dimensional space. Expressing space in terms of these vectors can done by matrices. Matrices is most advanced tool in mathematics to manipulate space. To understand them understanding of vectors is essential.
Matrices will be the our next topic. See you later…