4.1 — Position and displacement vectors

In 1D, position was a single number x. In 2D or 3D, we need a position vector that runs from the origin to where the object is.

Position vector \vec r

\[ \vec r = x\hat i + y\hat j + z\hat k \]

Interpretation: tail at origin, tip at the object's location.

Example: a particle at (3, 4, 5) m. Its position vector:

\[ \vec r = 3\hat i + 4\hat j + 5\hat k\ \mathrm{m} \]

Magnitude: r = \sqrt{9 + 16 + 25} = \sqrt{50} \approx 7.07 m.

Displacement vector \Delta \vec r

If an object moves from \vec r_1 to \vec r_2:

\[ \Delta \vec r = \vec r_2 - \vec r_1 = (x_2 - x_1)\hat i + (y_2 - y_1)\hat j + (z_2 - z_1)\hat k \]

Interpretation: the straight-line vector from start to end — path taken doesn't matter.

Example: a particle moves from (1, 2, 3) to (4, 6, 2). Displacement:

\[ \Delta \vec r = 3\hat i + 4\hat j - \hat k\ \mathrm{m} \]

Magnitude: |\Delta \vec r| = \sqrt{9 + 16 + 1} = \sqrt{26} \approx 5.1 m.

Distance vs displacement

As in 1D (§2.1), distance is the actual path length, displacement is straight-line from start to end.

Example: a particle goes from (0,0) to (3,0), then to (3,4).

• Distance: 3 + 4 = 7 m
• Displacement: \Delta \vec r = 3\hat i + 4\hat j, magnitude \sqrt{9+16} = 5 m

For very winding paths, distance can be much greater than displacement magnitude.

Trajectory — a time-parameterized vector

A moving object's position is time-dependent:

\[ \vec r(t) = x(t)\hat i + y(t)\hat j + z(t)\hat k \]

Each component is an independent function of time. The big key: 2D/3D motion decomposes into three independent 1D motions.

Example: a projectile in gravity:

• x(t) = v_{0x} t (uniform horizontal motion)
• y(t) = v_{0y} t - \frac{1}{2} g t^2 (vertical motion at constant acceleration)

Both look like Chapter 2 — just combined for the same particle.

Graphical view — the trajectory

Plotting \vec r(t) over time gives the trajectory — a curve in space.

Examples:

• Uniform straight-line motion: a straight line
• Projectile (no air drag): a parabola
• Circular motion: a circle
• Electron in a magnetic field: a helix

A few notes and common mistakes

1. Position vector depends on the origin. Shifting the origin changes \vec r but not \Delta \vec r.

2. Components can be negative. An object in the third quadrant (-5, -3) has \vec r = -5\hat i - 3\hat j. Signs matter.

3. Displacement can be zero but distance nonzero. An object that leaves and returns to the same point has \Delta \vec r = 0 but distance > 0.

What you should be able to do

• Write a position vector in component form
• Compute displacement by subtracting two positions
• Distinguish distance vs displacement
• Visualize the trajectory as the set of \vec r(t) points
• Recognize the decomposition of motion into independent components

Preview of §4.2

The derivative of \vec r(t) is the velocity vector. The second derivative is the acceleration vector. §4.2 is next.

📚 See also: Halliday Vol 1, Ch 4, §4.1 — Position and Displacement.

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