[Computer Animation] Vector Operations

Vector Operations

Scalar vs. Vector

• Scalar : a quantity, such as time or temperature, that has magnitude(규모) but not direction.
• Vector : a variable quantity, such as force, that has size and direction.

 

Scalar vs. Vector vs. Matrix

 

Addition and Scalar Multiplication

• Let $a$ and $b$ be 3D Vectors and $s$ is a scalar value.

• Vector Addition

• Vector-Scalar Multiplication

 

Vector Length (Magnitude)

 

Unit Vector

• A Unit Vector is any vector with a length of one(1).
• A Unit vector is often indicated with a hat as in $\hat{v}$
• Examples
  • $\hat{e} = (1, 0, 0)$
  • $\hat{a} = (\sqrt{\frac{1}{3}} , \sqrt{\frac{1}{3}}, \sqrt{\frac{1}{3}})$

 

Normalizing

• Normalizing a vector is to divide the vector by its length to create a unit vector.

 

Questions

 

Dot Product, $a · b$

• Sometimes called Inner Product or Scalar Product
• The result is Scalar

 

 

Questions

 

Cross Product, $a × b$

• Sometimes called Vector Product
• The result is new Vector
• The result vector is perpendicular(직각의) to both $a$ and $b$.

 

Questions

 

Normal Vector

• The Normal Vector to a surface is a vector which is perpendicular(직각의) to the surface at a given point.
• The length of a Normal Vector is always 1 (Unit Vector)

 

Question

 

Matrix-Vector Multiplication

 

Matrix-Matrix Multiplication

 

Summary