Foundations — Unit vectors — î, ĵ, k̂; constructing unit vector

Before you can read the parent note on unit vectors, you need every symbol it quietly assumes. Below, each item is built from the previous one: plain words → the picture → why the topic needs it. Nothing is used before it is drawn.

1. What is an arrow (a vector)?

Contrast this with an ordinary number like $5$ (a scalar) — that has size only, no direction. Temperature is a scalar; a shove is a vector (you need to know how hard AND which way).

Why the topic needs it: the whole idea of a "unit vector" only makes sense once you accept that an arrow secretly stores two independent facts. Look at the red arrow in the figure — its length and its tilt are two separate dials.

2. The number line, axes, and the origin

Why the topic needs it: to say "this arrow reaches $3$ to the right and $4$ up" you first need a right and an up to measure against. See Position vector and displacement for how points get their coordinates.

3. Components — splitting an arrow into "right" and "up" amounts

Look at the figure: drop a straight line from the arrow's tip straight down to the $x$-axis — the shadow it casts is $A_x$. Drop one straight across to the $y$-axis — that shadow is $A_y$. The arrow is the diagonal; its two shadows are the components.

Signs matter. If the arrow points left, $A_x$ is negative. If it points down, $A_y$ is negative. The sign of a component tells you which of two opposite directions along that axis. Never drop it — the parent note lists dropped signs as a top mistake. See Vectors — components and resolution for the full story.

4. The hat, and î, ĵ, k̂ — the pure pointers

Why the topic needs them: they are the alphabet of direction. Writing $\vec{A}=A_x\hat{i}+A_y\hat{j}$ means "go $A_x$ steps in the $\hat{i}$ direction, then $A_y$ steps in the $\hat{j}$ direction." The numbers carry amount; the hats carry direction. That clean split is exactly what the topic is built on.

5. Multiplying an arrow by a plain number (scaling)

Why the topic needs it: constructing a unit vector is scaling ($\hat{A}=\vec{A}\cdot \frac{1}{|\vec{A}|}$), and rebuilding a vector is scaling ($\vec{A}=|\vec{A}|\hat{A}$).

6. Magnitude and the square-root sign

To compute it we need two symbols:

• Squaring, $A_x^2=A_x\times A_x$. Squaring always gives something positive, which is why a negative component contributes the same as its positive twin — length has no sign.
• Square root, $\sqrt{\cdot }$, the undo of squaring: $\sqrt{25}=5$ because $5^2=25$.

Why a square root and not just adding the components? Because the components are perpendicular sides of a right triangle — and the long side of a right triangle is not the sum of the short sides. That is Pythagoras theorem:

For $\vec{A}=3\hat{i}+4\hat{j}$: plain addition gives the wrong $3+4=7$; Pythagoras gives the right $\sqrt{3^2+4^2}=\sqrt{25}=5$. See Magnitude and direction of a vector.

7. The degenerate case — the zero vector

How these foundations feed the topic

Read it top-down: arrows and axes give you components; components make a right triangle; Pythagoras turns that triangle into a magnitude; magnitude plus scaling plus the basis hats give you the unit vector.

Equipment checklist

Test yourself — you are ready when you can answer every line without peeking.

Connections

• Unit vectors — î, ĵ, k̂; constructing unit vector (the parent this page prepares you for)
• Vectors — components and resolution
• Magnitude and direction of a vector
• Pythagoras theorem
• Position vector and displacement