01-What exactly is a vector?#
Definition#
A vector is a quantity that has both magnitude and direction
Representation#
- Usually represented by arrows or directed line segments
- Key properties are length (magnitude) and direction
- Example: Vector (-2, 3) represents an arrow from origin (0,0) pointing to point (-2,3)
Vector magnitude calculation formula#
$$ |\vec{v}| = \sqrt{x^2 + y^2} $$
Characteristics#
- Multi-dimensional vectors exist (2D/3D most commonly used in gaming)
- Essentially an ordered list of numbers that can represent various things
Vector addition#
Can be viewed as moving from the origin along two vector directions, for example vector V + vector W means we first move in the direction of vector V, then move in the direction of vector W, with the movement distance being their magnitudes
Vector addition formula#
$$ \vec{a} + \vec{b} = \begin{bmatrix} x_1 \ y_1 \end{bmatrix} + \begin{bmatrix} x_2 \ y_2 \end{bmatrix} = \begin{bmatrix} x_1 + x_2 \ y_1 + y_2 \end{bmatrix} $$
Vector and scalar multiplication#
Also called vector scaling
Scalar multiplication formula#
$$ \vec{v} = k \cdot \vec{v} = (k v_x, k v_y, k v_z) $$
02-Linear combinations, span, and basis#
Basis vectors#
In a 2D coordinate system (xy plane), there are two special vectors: one pointing to the positive right with length 1, called “i-hat”, and another pointing straight up with length 1, called “j-hat”. These two vectors together form the “basis vectors” of this coordinate system, they are the foundational framework of the coordinate system.
For example, vector (3, -2) can be understood as:
- Stretching the basis vector i-hat by 3 times
- Stretching the basis vector j-hat by 2 times
- The final vector (3, -2) is the sum of these two scaled vectors (3i + (-2j))
Here, the coordinate values 3 and -2 are treated as scalars, while the basis vectors are the objects being scaled by these scalars
Linear combination#
The sum of two vector scalar multiplications is called a linear combination
$$ \vec{U} = a\vec{V} + b\vec{W} $$
where a,b are scalars
Span#
Imagine you have several arrows, the span is the set of all possible positions you can reach using only these arrows through the two most fundamental operations: vector addition and scalar multiplication
One-dimensional#
Collinear means the vectors that constitute the space are on the same line
Two-dimensional#
Three-dimensional#
There exists a vector that is not in the same plane as other vectors. When you scale this vector, it moves the plane along its direction, thus reaching any point in 3D space
Linear dependence#
If removing one vector doesn’t make the span smaller, then this vector is redundant. It itself is obtained from other vectors through scalar multiplication and addition. Such vectors are called linearly dependent
Linear independence#
If removing one vector makes the span smaller, it means this arrow is not redundant. Such vectors are called linearly independent
Basis#
A basis of a vector space is a minimal set of linearly independent vectors
- Can span the entire space
- Are all linearly independent