Foundations — Projection of vectors

This page assumes you have seen nothing. We will name every symbol the parent Projection of vectors note uses, draw the picture behind it, and say why the topic cannot live without it. Read top to bottom; each block earns the next.

1 — What is a vector? The symbol $\vec{a}$

A plain number like $5$ has only size. An arrow has size and which way it points. That "which way" is the whole reason projection exists — we are comparing directions.

In the picture the burnt-orange arrow starts at the origin (the corner point $O$) and ends at its tip. The length of the arrow is how big it is; the tilt of the arrow is its direction.

2 — Coordinates: writing $\vec{a}=(3,4)$

• $\vec{a}=(3,4)$ means: from $O$, step 3 to the right, then 4 up. The arrow to that landing spot is $\vec{a}$.
• A three-dimensional vector just adds a third number for "depth": $\vec{a}=(2,1,2)$ (used in Example 4 of the parent).

3 — Length of a vector: the symbol $|\vec{a}|$

Where does its formula come from? Look again at figure s02: the components $a_x$ and $a_y$ form the two short sides of a right triangle, and the arrow $\vec{a}$ is the slanted long side (the hypotenuse). Pythagoras — "the two legs squared, added, then square-rooted" — gives the length:

$|\vec{a}|=\sqrt{a_x^2+a_y^2}$

Why the topic needs it. The scalar projection $|\vec{a}|\cos \theta$ literally begins with $|\vec{a}|$ — you cannot talk about "how far the shadow reaches" without a notion of length. And division by $|\vec{b}|$ in the formula is what stops the answer growing just because you drew $\vec{b}$ longer.

4 — The angle between two arrows: $\theta$

The figure shows three cases you MUST be ready for:

• Acute ($\theta <90^{\circ }$): arrows roughly agree in direction. Shadow lands forward.
• Right angle ($\theta =90^{\circ }$): arrows are perpendicular. Shadow has zero length — no part of $\vec{a}$ points along $\vec{b}$.
• Obtuse ($\theta >90^{\circ }$): arrows disagree, pointing apart. Shadow falls on the back side — this is where the parent's minus sign comes from.

Keep all three in mind: the parent's "sign matters" section is nothing more than these three pictures.

5 — Cosine: why $\cos \theta$ and not some other function

Sign of cosine — covering every case:

• $\cos \theta >0$ for $\theta <90^{\circ }$ → shadow forward (positive).
• $\cos \theta =0$ at $\theta =90^{\circ }$ → shadow zero.
• $\cos \theta <0$ for $\theta >90^{\circ }$ → shadow backward (negative).

So the single number $\cos \theta$ carries both how much and which side — that's why the parent insists you never take its absolute value.

6 — The dot product: the symbol $\vec{a}\cdot \vec{b}$

7 — Unit vectors: the symbol $\hat{b}$

This is also why the vector projection divides by $|\vec{b}|$ twice: once inside $\hat{b}$ to normalise the direction, once to get the length. That doubled length is written $\vec{b}\cdot \vec{b}=|\vec{b}{|}^2$.

8 — Perpendicular / orthogonal: the symbol ${\vec{a}}_{\perp }$ and "$\cdot =0$"

Why does zero mean perpendicular? From the geometry form $\vec{a}\cdot \vec{b}=|\vec{a}||\vec{b}|\cos 90^{\circ }$, and $\cos 90^{\circ }=0$. So the dot product vanishes precisely when they meet at a right angle. The symbol ${\vec{a}}_{\perp }$ (read "a-perp") is the leftover part of $\vec{a}$ once you remove its shadow — it points at a clean right angle to $\vec{b}$. This is the seed of Orthogonal decomposition.

Prerequisite map

Equipment checklist

Test yourself — each line is prompt ::: answer.

Connections

• Dot product — the number-machine that hides $\cos \theta$; the reason projection needs no protractor.
• Unit vectors — supplies the pure direction $\hat{b}$ that carries the shadow.
• Orthogonal decomposition — where ${\vec{a}}_{\perp }$ leads next.
• Projection of vectors — the parent topic these foundations feed.