Vectors in ℝⁿ — operations, geometric interpretation: Foundations

This page assumes nothing. Every squiggle, letter, and picture the parent note used is built here from the ground up, in an order where each brick rests on the one before it.

0 — What is a number line, and what is a "real number"?

Before arrows, we need the ground they live on.

We need $R$ because every coordinate of every vector is one of these ruler-points.

1 — From one ruler to $n$ rulers: the symbol $R^{n}$

One ruler locates you on a line. To locate yourself on a map you need two rulers (east–west, north–south). To locate a point in a room you need three (add up–down).

Figure — Vectors in ℝⁿ — operations, geometric interpretation

2 — The vector itself: bold $v$ , components $v_{i}$ , and the subscript

Column notation. The parent writes vectors stacked vertically: $v = v_{1} ⋮ v_{n} .$ This is the same list, just written top-to-bottom instead of left-to-right. The tall square brackets are a container; the vertical dots $⋮$ again mean "continue the pattern." Column form is pure convention — it will matter later when matrices multiply vectors, but for now $(v_{1}, \dots, v_{n})$ and the column are identical.

3 — The two engines: $+$ and scalar multiplication

Everything the parent builds runs on exactly two moves.

4 — Length: the square-root sign and $∥ v ∥$

To ask "how long is this arrow?" the parent invokes Pythagoras. Two symbols show up.

5 — The dot product: $\cdot$ , $θ$ , $cos$ , $⊥$

The parent's climax turns arithmetic into angles. Four new symbols.

Recall Quick self-check: which symbol returns what?

Does $∥ v ∥$ give a vector or a number? ::: A single non-negative number (a length). Does $u \cdot v$ give a vector or a number? ::: A single number (scalar) — never an arrow. Does $c v$ give a vector or a number? ::: A vector (a re-stretched arrow). What is $v \cdot v$ ? ::: $∥ v ∥^{2}$ — the squared length.

How these foundations feed the topic

This map is the parent note in miniature: real numbers build $R^{n}$ , which houses vectors, whose dual picture powers the two engines, from which length and the dot product — and finally angle and perpendicularity — flow. Deeper branches live in Linear Combinations and Span, Dot Product and Orthogonality, and Norms and Distance in Rn, and the whole structure is formalized in Vector Spaces — Axioms.

Equipment checklist

Self-test: can you say each in one plain sentence before revealing?

R

The set of all real numbers — every point on an endless ruler.

R^{n}

All lists of

n

real numbers;

n

independent perpendicular rulers/axes (superscript counts axes, not a power).

\in

"is an element of / lives inside," e.g.

v_{i} \in R

v

(bold)

A vector — one list from

R^{n}

, read as both a point and an arrow from the origin.

v_{i}

and subscript

i

The

i

-th component; the subscript is an address pointing to entry number

i

0

The origin / zero vector, all components zero — where arrows are rooted.

scalar

c

A single real number used to stretch/shrink a vector.

component-wise

Do the operation one matching entry at a time.

c v

Scaling: stretch length by

∣ c ∣

, flip if

c < 0

, collapse to

0

c = 0

∣ c ∣

Absolute value — distance of

c

from

0

, always non-negative.

v_{i}^{2}

and

Squaring (area of a side-

v_{i}

square, kills sign) and square root (its undo).

∥ v ∥

The norm — the arrow's straight-line length, a non-negative number.

\sum_{i = 1}^{n}

Summation loop: add the terms as

i

runs from

1

n

\hat{v}

Unit vector —

v

shrunk to length

1

, same direction, via

v /∥ v ∥

u \cdot v

Dot product — sum of products of matching components; a single scalar.

θ

The angle between two arrows sharing a tail.

cos θ

Alignment dial from

- 1

+ 1

1

same way,

0

right angle,

- 1

opposite.

⊥

Perpendicular (right angle) — holds exactly when the dot product is

0

⟺

"If and only if" — both sides true together, a two-way implication.