Matrix Rotations

James Blackburn

January 2022

1 Using the angle between two vectors

First of all, we can define the 2x2 matrix inversion in the variable R as:

Where θ is the anti- clockwise angle of rotation. [cosθ - sinθ] R = sinθ cos θ
(1)

(2)
The general form of vectors is:
[ ] x ⃗v = y
(3)
Now let’s take the identity matrix:
[ ] I = 1 0 0 1
(4)
Now, let’s take the vector of the positive x-axis and the positive y-axis:
[ ] ⃗a = 1 0 ⃗ [0] b = 1
(5)

(6)
And rotate this by the matrix R:
[ ][ ] [ ] ⃗c = cosθ - sinθ 1 = cosθ sin θ cosθ 0 sin θ ⃗ [cosθ - sinθ] [0] [- sin θ] d = sinθ cosθ 1 = cosθ
(7)

(8)

Using this function, we can give evidence of the dot product between two vectors.
The dot product is defined as such (where a and b are random variables):

cosθ = xa ×-xb +-ya ×-yb |a|× |b|
This can be re-arranged to find θ (theta):
( ) θ = cos-1 xa ×-xb +-ya ×-yb |⃗a|× |⃗b|
Now let’s find the rotation between ⃗a & ⃗c and ⃗b & ⃗d :
( ) -1 -----1×-cosθ-+-0×-sinθ------ θ1 = cos √12-+-02 × ∘ (cos-θ)2 +-(sin-θ)2 ( ) = cos-1 ----------c∘osθ---------- √12-+-02 × cos2 θ+ sin2θ (cosθ ) = cos-1 ----- 1 ×1 = cos- 1(cosθ) = θ ( ) θ2 = cos-1 √----0-×--s∘inθ+-1-×cosθ------ 02 + 12 × (- sinθ)2 + (cosθ)2 ( cosθ ) = cos-1 √---------∘------------- 02 + 12 × sin2θ +( cos2θ ) -1 cosθ- = cos 1× 1 = cos-1 (cosθ) = θ
(9)

(10)
Therefore, we have proved that the rotation matrix R rotates a vector by the angle θ.

Alongside this proof, we have also managed to prove another fact: Given any 2d matrix, one can instantly tell if it’s a rotation matrix by the following test:

[ ] A = a b ∘ ----c- d a2 + c2 = 1 ∘ -2---2 b + d = 1
(11)

(12)

(13)
Because:
∘ ------------ ∘ cos2θ+-sin2θ-= 1 sin2θ+ cos2θ = 1
(14)

(15)

2 Clockwise or anti-clockwise

Considering that a rotation is always going to be in the range 0 ≤ θ ≤ 360 and considering the 3 functions we use in the matrix R are sinθ, cosθ and -sinθ, we can plot these functions onto a graph:

sin θ cosθ 0611233--001θf028406 1 0.5(00000.θ5) -sinθ

Figure 1: Domain: 0 ≤ θ ≤ 360, Range: -1 ≤ f(θ) ≤ 1

Now, let’s consider what an anti-clockwise and clockwise rotation actually is. Because we know that the matrix R gives an anti-clockwise rotation, we know that 0 ≤ θ ≤ 180 is anti-clockwise, hence 180 ≤ θ ≤ 360 is clockwise.

Let’s zoom in to the cosθ graph:

cosθ acw 0611233--001θf028406 1 0.5(θ00000.)5 cosθ cw

Figure 2: Domain: 0 ≤ θ ≤ 360, Range: -1 ≤ f(θ) ≤ 1
Can you see how it doesn’t matter whether the rotation is clockwise or anti-clockwise? It is negative or positive in either of these cases. This means we can forget about the cosine graph for now! Let’s move onto sinθ and -sinθ!

sin θ 0611233--001θf028406 1 0.5(θ00000.)5 -sinθ

Figure 3: Domain: 0 ≤ θ ≤ 360, Range: -1 ≤ f(θ) ≤ 1
Can you see how the graphs converge at 180o? This means that in the domain 0 ≤ θ ≤ 180 (anti-clockwise), 0 ≤ sinθ ≤ 1 and -1 ≤-sinθ ≤ 0. Hence, for the domain 180 ≤ θ ≤ 360 (clockwise), -1 ≤ sinθ ≤ 0 and 0 ≤-sinθ ≤-1.

But what does this actually mean for a mathematician? Well, it means, given the following rotation matrix:

⌊ √-⌋ 1 - -3- B = ⌈ 2√- 2 ⌉ 23- 12

We can see that B12 (the top right corner) is negative. Therefore, it is an anticlockwise rotation!

Now let’s take another matrix C:

[ 1-- 1-] √2 √2 C = -√1- 1√-- 2 2

We can see that C21 (the bottom left corner) is negative. Therefore, it is a clockwise rotation!