Displacement and velocity vectors are key concepts in understanding motion in multiple dimensions. They help us describe an object's position, movement, and speed in space using mathematical tools.

These vectors allow us to analyze complex motions, from a car's journey to a spacecraft's trajectory. By breaking down movement into components, we can tackle real-world problems and predict object behavior in various scenarios.

Displacement and Velocity Vectors

Position vectors for multidimensional displacements

• Represent the location of an object in space relative to a chosen origin (reference point)
• In 2D, a position vector is expressed as $\vec{r} = x\hat{i} + y\hat{j}$
  • $x$ and $y$ represent the scalar components of the vector along the x and y axes
  • $\hat{i}$ and $\hat{j}$ denote the unit vectors in the x and y directions (basis vectors)
• In 3D, a position vector is expressed as $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$
  • $x$, $y$, and $z$ represent the scalar components of the vector along the x, y, and z axes
  • $\hat{i}$, $\hat{j}$, and $\hat{k}$ denote the unit vectors in the x, y, and z directions (basis vectors)
• Examples:
  • 2D: A person standing 3 m east and 4 m north of a reference point has a position vector $\vec{r} = 3\hat{i} + 4\hat{j}$
  • 3D: A drone flying 5 m east, 2 m north, and 10 m above a reference point has a position vector $\vec{r} = 5\hat{i} + 2\hat{j} + 10\hat{k}$

Displacement in two and three dimensions

• Displacement represents the change in position of an object
• Displacement is a vector quantity, denoted by $\Delta\vec{r}$
• In 2D, displacement is calculated as $\Delta\vec{r} = \Delta x\hat{i} + \Delta y\hat{j}$
  • $\Delta x$ and $\Delta y$ represent the changes in the x and y components of the position vector
• In 3D, displacement is calculated as $\Delta\vec{r} = \Delta x\hat{i} + \Delta y\hat{j} + \Delta z\hat{k}$
  • $\Delta x$, $\Delta y$, and $\Delta z$ represent the changes in the x, y, and z components of the position vector
• Examples:
  • 2D: A car moves from point A (2 m, 3 m) to point B (5 m, 7 m), its displacement is $\Delta\vec{r} = (5-2)\hat{i} + (7-3)\hat{j} = 3\hat{i} + 4\hat{j}$
  • 3D: A bird flies from a branch (1 m, 2 m, 5 m) to a nest (4 m, 6 m, 8 m), its displacement is $\Delta\vec{r} = (4-1)\hat{i} + (6-2)\hat{j} + (8-5)\hat{k} = 3\hat{i} + 4\hat{j} + 3\hat{k}$
• Displacement can be visualized using a displacement-time graph, which shows how displacement changes over time

Velocity vectors from position vectors

• Velocity represents the rate of change of position with respect to time
• Velocity is a vector quantity, denoted by $\vec{v}$
• Instantaneous velocity is the limit of average velocity as the time interval approaches zero
  • Instantaneous velocity is the derivative of the position vector with respect to time: $\vec{v} = \frac{d\vec{r}}{dt}$
• In 2D, the velocity vector is $\vec{v} = v_x\hat{i} + v_y\hat{j}$
  • $v_x = \frac{dx}{dt}$ and $v_y = \frac{dy}{dt}$ represent the scalar components of velocity in the x and y directions
• In 3D, the velocity vector is $\vec{v} = v_x\hat{i} + v_y\hat{j} + v_z\hat{k}$
  • $v_x = \frac{dx}{dt}$, $v_y = \frac{dy}{dt}$, and $v_z = \frac{dz}{dt}$ represent the scalar components of velocity in the x, y, and z directions
• Examples:
  • 2D: If a particle's position is given by $\vec{r}(t) = (3t)\hat{i} + (2t^2)\hat{j}$, its velocity is $\vec{v}(t) = \frac{d\vec{r}}{dt} = 3\hat{i} + (4t)\hat{j}$
  • 3D: If an object's position is given by $\vec{r}(t) = (2t^2)\hat{i} + (3t)\hat{j} + (5t^3)\hat{k}$, its velocity is $\vec{v}(t) = \frac{d\vec{r}}{dt} = (4t)\hat{i} + 3\hat{j} + (15t^2)\hat{k}$

Average velocity across multiple dimensions

• Average velocity represents the displacement divided by the time interval
• Average velocity is a vector quantity, denoted by $\vec{v}_{avg}$
• In 2D, average velocity is calculated as $\vec{v}_{avg} = \frac{\Delta\vec{r}}{\Delta t} = \frac{\Delta x\hat{i} + \Delta y\hat{j}}{\Delta t}$
  • $\Delta x$ and $\Delta y$ represent the changes in the x and y components of the position vector
  • $\Delta t$ represents the time interval
• In 3D, average velocity is calculated as $\vec{v}_{avg} = \frac{\Delta\vec{r}}{\Delta t} = \frac{\Delta x\hat{i} + \Delta y\hat{j} + \Delta z\hat{k}}{\Delta t}$
  • $\Delta x$, $\Delta y$, and $\Delta z$ represent the changes in the x, y, and z components of the position vector
  • $\Delta t$ represents the time interval
• Examples:
  • 2D: A cyclist rides from point A (0 m, 0 m) to point B (30 m, 40 m) in 10 s, their average velocity is $\vec{v}_{avg} = \frac{(30-0)\hat{i} + (40-0)\hat{j}}{10} = 3\hat{i} + 4\hat{j}$ m/s
  • 3D: A spacecraft travels from Earth (0 m, 0 m, 0 m) to Mars (2.2×10^11 m, 0 m, 0 m) in 180 days, its average velocity is $\vec{v}_{avg} = \frac{(2.2×10^11-0)\hat{i} + (0-0)\hat{j} + (0-0)\hat{k}}{180×24×3600} ≈ 14,120\hat{i}$ m/s

Motion and Reference Frames

• Relative motion describes the movement of an object with respect to another moving object or frame of reference
• A frame of reference is a coordinate system used to describe the position and motion of objects
• The trajectory of an object is the path it follows through space over time
• Path length refers to the total distance traveled by an object along its trajectory, which may differ from its displacement