Components and unit vectors

The graphical method (§3.2) is fine but doesn’t scale. Components are the algebraic tool that makes vector work scalable — 100 vectors as easy as 2. Each vector is completely described by a few scalar numbers (its components).

Decomposition — one axis at a time

Take a Cartesian coordinate system (x, y in 2D). A vector \vec a decomposes into two sub-vectors:

• x-component: \vec a_x — projection of \vec a onto the x axis
• y-component: \vec a_y — projection onto the y axis

$$
\vec a = \vec a_x + \vec a_y
$$

Each component is a signed scalar:

• a_x = length of \vec a_x (positive if in +x direction, negative if -x)
• a_y = same for y

Trig formulas

If \vec a has magnitude a = |\vec a| and angle \theta from the x axis:

$$
a_x = a \cos\theta
$$

$$
a_y = a \sin\theta
$$

And the reverse — given components:

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

$$
\tan\theta = \frac{a_y}{a_x}
$$

Unit vectors — \hat i, \hat j, \hat k

A unit vector is a vector of magnitude 1 carrying only direction. The three standard Cartesian unit vectors:

• \hat i (or \hat x): unit vector in the +x direction
• \hat j (or \hat y): unit vector in the +y direction
• \hat k (or \hat z): unit vector in the +z direction (3D)

Properties:
– |\hat i| = |\hat j| = |\hat k| = 1
– \hat i \perp \hat j \perp \hat k (mutually perpendicular)

Full vector in unit-vector notation

Any vector can be written as a linear combination of unit vectors:

$$
\vec a = a_x \hat i + a_y \hat j\quad (\text{in 2D})
$$

$$
\vec a = a_x \hat i + a_y \hat j + a_z \hat k\quad (\text{in 3D})
$$

This notation is powerful — every vector operation reduces to simple algebra of components.

Numerical example

Scenario: a displacement \vec d = 10 m at \theta = 30° from the x axis.

Components:

$$
d_x = 10 \cos(30°) = 10 \cdot 0.866 \approx 8.66\ \mathrm{m}
$$

$$
d_y = 10 \sin(30°) = 10 \cdot 0.5 = 5.00\ \mathrm{m}
$$

Unit-vector notation:

$$
\vec d = 8.66\, \hat i + 5.00\, \hat j\ \mathrm{m}
$$

Reverse check:

$$
d = \sqrt{8.66^2 + 5.00^2} = \sqrt{75 + 25} = \sqrt{100} = 10\ \mathrm{m}\ ✓
$$

Unit vector in an arbitrary direction

Given \vec a, the unit vector in the same direction:

$$
\hat a = \frac{\vec a}{|\vec a|} = \frac{\vec a}{a}
$$

Useful whenever only direction matters, not magnitude — e.g., defining an electric field from a point charge.

Signs and quadrants

Depending on which quadrant the vector lies in:

Quadrant	\theta	a_x	a_y
I	0° \leq \theta < 90°	+	+
II	90° \leq \theta < 180°	-	+
III	180° \leq \theta < 270°	-	-
IV	270° \leq \theta < 360°	+	-

Warning: when computing \theta from \tan\theta = a_y/a_x, calculators only return quadrants I and IV. For II and III, add 180°. Always cross-check with the signs of a_x and a_y.

A few notes and common mistakes

1. A component is a scalar, not a vector.
a_x is a signed number. \vec a_x = a_x \hat i is a vector. Keep the distinction.

2. Angles are always from a specified reference axis.
Usually from +x, positive counterclockwise. Be consistent.

3. For a 3D vector, don’t use three angles.
Use two angles (like \theta, \phi in spherical coords) or three components (x, y, z). Three angles are redundant and confusing.

4. If a component is negative, that’s fine.
Don’t try to fix the sign manually — use \theta; cos and sin get the signs right on their own.

What you should be able to do

After this section, you should be able to:

• Decompose a vector into a_x, a_y given magnitude and \theta
• Do the reverse — from components to magnitude and angle (Pythagoras + arctan)
• Recognize the unit vectors \hat i, \hat j, \hat k
• Write a vector as a_x \hat i + a_y \hat j + a_z \hat k
• Compute the unit vector in a given direction via \hat a = \vec a / |\vec a|
• Identify the correct quadrant from component signs

Preview of §3.4

With components in hand, vector addition becomes trivial: add corresponding components. §3.4 shows the tool with real-physics examples.

📚 See also: Halliday Vol 1, Ch 3, §3.3 — Vector Components.
📖 Open reference: OpenStax University Physics Vol 1 — Chapter 2.2.