Vectors, Displacement, and Velocity in Physics

Vectors and Their Operations

Vector Multiplication by a Scalar

Multiplying a vector by a scalar changes the magnitude of the vector but not its direction (unless the scalar is negative, which reverses the direction).

• Definition: If \( \vec{A} \) is a vector and \( \lambda \) is a scalar, then \( \lambda \vec{A} \) is a vector with magnitude \( |\lambda| \cdot |\vec{A}| \) and direction same as \( \vec{A} \) (or opposite if \( \lambda < 0 \)).

• Example: If \( \vec{A} = 3\hat{i} + 4\hat{j} \) and \( \lambda = 2 \), then \( 2\vec{A} = 6\hat{i} + 8\hat{j} \).

Vector Subtraction

Subtracting one vector from another gives a third vector that points from the tip of the second vector to the tip of the first.

• Definition: \( \vec{C} = \vec{A} - \vec{B} \) is equivalent to \( \vec{C} = \vec{A} + (-\vec{B}) \).

• Geometric Interpretation: To subtract \( \vec{B} \) from \( \vec{A} \), reverse the direction of \( \vec{B} \) and add it to \( \vec{A} \).

• Diagram: In the provided notes, vector \( \vec{C} \) points from the tip of \( \vec{B} \) to the tip of \( \vec{A} \).

• Example: If \( \vec{A} = 5\hat{i} \) and \( \vec{B} = 2\hat{i} \), then \( \vec{A} - \vec{B} = 3\hat{i} \).

Displacement

Definition and Formula

Displacement is the change in position of an object. It is a vector quantity, having both magnitude and direction.

• Formula:

$\Delta \vec{r} = \vec{r}_{\text{fin}} - \vec{r}_{\text{ini}}$

• \( \vec{r}_{\text{fin}} \): Final position vector

• \( \vec{r}_{\text{ini}} \): Initial position vector

• Example: If an object moves from \( (2, 3) \) to \( (5, 7) \), then \( \Delta \vec{r} = (5-2)\hat{i} + (7-3)\hat{j} = 3\hat{i} + 4\hat{j} \).

Velocity

Definition and Formula

Velocity is the rate of change of displacement with respect to time. It is a vector quantity, indicating both speed and direction.

• Formula:

$\vec{v} = \frac{\Delta \vec{r}}{\Delta t}$

• \( \Delta \vec{r} \): Displacement vector

• \( \Delta t \): Time interval

• Units: Meters per second (m/s) in SI units

• Example: If an object moves 10 meters east in 2 seconds, \( \vec{v} = \frac{10\hat{i}}{2} = 5\hat{i} \; \text{m/s} \).

Additional info: In physics, velocity differs from speed in that it includes direction, while speed is the magnitude of velocity.