Foundations — Resolution of vectors — into components (any axes)

Before you can resolve a single vector, the parent note quietly used a whole toolbox: arrows, angles, right triangles, sine, cosine, unit vectors, dot products. This page builds every one of those from zero, in the order they depend on each other. If a symbol appears in the parent note, it is defined here first.

1. The arrow: what a vector is

Think of pushing a toy car. Saying "I pushed with 5 units of force" is not enough — which way did you push? A vector answers both at once.

• We write a vector as $\vec{A}$ — the little arrow on top means "this is a vector, not just a plain number."
• A plain number (like temperature, or just a length) with no direction is called a scalar.

2. Magnitude: the length of the arrow

Picture the arrow lying on graph paper. Take a ruler. The number you read is $A$. It is never negative — a length cannot be less than nothing.

Why the topic needs it: the parent's key formulas $A_x=A\cos \theta$ and $A_y=A\sin \theta$ multiply this length $A$ by a fraction. Without a length to scale, there is nothing to resolve.

3. The angle $\theta$: which way the arrow tilts

• $\theta =0^{\circ }$ means the arrow points straight right.
• $\theta =90^{\circ }$ means straight up.
• The little ${}^{\circ }$ symbol means "degrees" — a full turn around a point is $360^{\circ }$.

Why the topic needs it: $\theta$ is the single number that tells us how the arrow is tilted, and therefore how much of it "leaks" sideways versus upward.

4. The right triangle: the secret shape inside every vector

The three sides each have a name relative to the angle $\theta$:

• Hypotenuse — the longest side, always opposite the square corner. Here it is the arrow itself, length $A$.
• Adjacent — the side next to (touching) the angle $\theta$, lying flat along the x-axis. This will become $A_x$.
• Opposite — the side across from $\theta$, standing up vertically. This will become $A_y$.

Why the topic needs it: resolution is reading the two shorter sides of this triangle. Every formula below is a ratio of these three sides.

5. Sine and cosine: rules that turn an angle into a ratio

We now need a tool that answers: "given the tilt $\theta$, what fraction of the arrow lies flat, and what fraction stands up?" Sine and cosine are exactly those two fractions.

Rearranging the definitions gives the parent's core result directly. Since $\cos \theta =A_x/A$, multiply both sides by $A$: $A_x=A\cos \theta ,A_y=A\sin \theta .$

6. Components $A_x$ and $A_y$: the two hidden arrows

• The subscript $x$ in $A_x$ just labels which direction — "the part of A along x."
• A component can be negative: if the arrow points left, $A_x$ is negative, meaning "reaching in the minus-x direction."

Why the topic needs it: these two numbers are the whole point of resolution. Once you have them, you can add vectors by adding numbers.

7. Pythagoras: rebuilding the arrow from its pieces

If we start from the two components and want the original length back, we use the rule connecting the three sides of a right triangle.

The little $2$ exponent means "multiply by itself" ($A_x^2=A_x\times A_x$). The  ​ symbol undoes squaring — it asks "what number, squared, gives this?"

Why the topic needs it: this is the inverse direction of resolution — going from two-numbers back to length-and-angle. The parent calls it "recovering magnitude."

8. Tangent and arctangent: recovering the angle

To get $\theta$ back from the components we need a tool relating the two short sides to each other.

To then extract $\theta$ itself we ask the reverse question, "which angle has this tangent?" — that reverse operation is written ${\tan }^{-1}$ (read "arctangent" or "inverse tan"): $\theta ={\tan }^{-1}\left(\frac{A_y}{A_x}\right).$

The small ${}^{-1}$ here does not mean "one over" — it means "undo." ${\tan }^{-1}$ takes a ratio and hands back the angle that produced it.

9. Unit vectors $\hat{x}$, $\hat{y}$, $\hat{n}$: arrows of length exactly one

Think of a unit vector as a signpost: it carries no size, only a direction. Multiplying it by a number scales it to any length.

Why the topic needs it: $\hat{n}$ lets the parent generalise from x–y axes to any axis. To find the shadow along a slanted direction, you first name that direction with a unit vector.

See Unit vectors and Cartesian coordinates for the full treatment.

10. The dot product: measuring one arrow's shadow on another

The parent's final tool answers: "how much of $\vec{A}$ points along the direction $\hat{n}$?"

The dot ($\cdot$) is the symbol for this operation; it always eats two vectors and gives back one plain number (a scalar). Notice it reuses cosine — because a shadow is again "adjacent-over-hypotenuse," just measured along a tilted direction instead of the x-axis.

Why the topic needs it: this is resolution onto any axis, not just horizontal/vertical. Full details in Dot product & scalar projection.

Prerequisite map

Every box on the left must be solid before the topic on the right makes sense. Follow the arrows and you have built the whole toolbox from zero.

Equipment checklist

Connections

• Parent: Resolution of vectors — into components (any axes)
• Unit vectors and Cartesian coordinates
• Dot product & scalar projection
• Vectors — addition (parallelogram & triangle law)