A bit of Trigonometry

Right angle triangle

• sin(θ) = Opposite / Hypotenuse
• cos(θ) = Adjacent / Hypotenuse
• tan(θ) = Opposite / Adjacent
• Pythagoras Theorem: \(a^2 + b^2 = c^2\)

Other triangle

• The Law of Cosines: \(c^2=a^2+b^2\)
• The Law of Sines: \(\frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}\)

Vector Magnitude and Direction

• Magnitude of u (distance): \(\|\vec{u}\|=\sqrt{\|u_1\|^2+\|u_2\|^2}\) (ie. Pythagorean Theorem)
• Direction vector of u(u1, u2) is a unit vector with the same direction: \(\vec{w}=(\frac{u1}{\|u\|},\frac{u2}{\|u\|})\)
• For right triangle, you also have the following properties: \(cos(\theta)=\frac{u_1}{\|u\|}; cos(\alpha)=\frac{u_2}{\|u\|}\)

Dot product of a and b = scalar

• Dot product of x and y:
  • \(\vec{x}\cdot\vec{y}=\|x\|\|y\|cos\theta=x_1y_1+x_2y_2\)
• It involves multiplying two vectors together to get a scalar, not another vector (for this reason, the dot product is often referred to as a scalar product).
• Two vectors a and b are orthogonal if and only if \(\vec{a}\cdot\vec{b} = 0 \)

Cross product of a and b = another vector

• it multiplies two vectors together to obtain a third vector rather than a scalar.
• a×b is a vector that is perpendicular to both a and b.

Orthogonal projection of a vector onto another

• By definition: \(Proj_y \vec{x} = \vec{z} = (\frac{\vec{x}\cdot\vec{y}}{\|y\|^2})\vec{y}\) (ie. a scalar times vector y)
• For better understand the orthogonal projection, you can watch this video
• Lets prove the formula above:

1. \(cos\theta = \frac{\|\vec{z}\|}{\|\vec{x}\|}\) (for right angle trigonometry)
2. \(cos\theta = \frac{\vec{x}\cdot\vec{y}}{\|\vec{x}\|\|\vec{y}\|}\) (for dot product)
3. Then, \(\frac{\|\vec{z}\|}{\|\vec{x}\|} = \frac{\vec{x}\cdot\vec{y}}{\|\vec{x}\|\|\vec{y}\|}\)
4. \(\|z\| = \frac{\vec{x}\cdot\vec{y}}{\|\vec{y}\|}\) (Multiply \(\|\vec{x}\|\) on each side)
5. Since vector z and y are at the same direction, they will have the same direction vector. \(\frac{\vec{z}}{\|\vec{z}\|}=\frac{\vec{y}}{\|\vec{y}\|}\).
6. Then, we will have \(\vec{z} = (\frac{\vec{x}\cdot\vec{y}}{\|\vec{y}\|^2}) \vec{y}\)