Linear_Algebra // Mod_01

VECTORS

The fundamental building blocks of space. Bridging the gap between raw lists of numbers and physical arrows in geometry.

01 // The Physics

In computer science, a vector is just an array: a list of numbers like [3,2][3, 2]. In physics, a vector is an arrow pointing in space. Linear algebra allows us to seamlessly translate between these two realities.

Magnitude
The Length. "How much?" Denoted as ∣∥v⃗∥∥|\|\vec{v}\|\|.
Direction
The Angle. "Which way?" It establishes trajectory.
Direction AND Magnitude!

Parallelogram Rule

Vector v[3, 2]
Vector w[1, 4]
Resultant Vector (v + w)
[3, 2]+[1, 4]=[4, 6]
v⃗\vec{v}
w⃗\vec{w}
v⃗+w⃗\vec{v} + \vec{w}

02 // The Algebra of Space

Addition
Tip-to-Tail

To add vectors, simply add their corresponding coordinates. Geometrically, it means walking along the first vector, then starting the second vector from where you stopped.

v⃗+w⃗=[v1+w1v2+w2]\vec{v} + \vec{w} = \begin{bmatrix} v_1+w_1 \\ v_2+w_2 \end{bmatrix}
Scaling
Scalar Mult

Multiplying a vector by a scalar (a normal number) stretches or shrinks it. Multiplying by a negative number flips it exactly 180°.

c⋅v⃗=[cv1cv2]c \cdot \vec{v} = \begin{bmatrix} cv_1 \\ cv_2 \end{bmatrix}
Dot Product
Alignment

A single number that tells you how much two vectors point in the same direction. If they are perpendicular, their dot product is exactly zero.

v⃗⋅w⃗=v1w1+v2w2\vec{v} \cdot \vec{w} = v_1 w_1 + v_2 w_2

Vectors Constructed

You are ready to pack them together into Grids.

Next: Matrices