Vectors in Physics

Definition and Properties of Vectors

Vectors are quantities that have both magnitude and direction. They are fundamental in physics for describing motion, forces, and many other physical phenomena.

• Component of a Vector: The projection of a vector along the axes (x and y), found by dropping perpendiculars from the vector to the axes.

• Resultant Vector: The sum of two or more vectors.

• Oblique Vector: A vector that does not lie on an axis.

• Magnitude of a Vector: The length or size of the vector, denoted as .

• Commutativity: Vector addition is commutative: .

Vector Components and Magnitude

• Given a vector with components and :

  • (angle with respect to the x-axis)

• Example: Walking 4 blocks east and 3 blocks north forms a right triangle. The resultant displacement is $5 north of east.

Vector Addition (Graphical and Analytical)

• Graphical Method: Vectors are added using the head-to-tail method.

• Analytical Method: Add corresponding components:

  • ,

• Example: If and , then .

One- and Two-Dimensional Kinematics

Position, Displacement, Velocity, and Acceleration Vectors

Describing motion in two dimensions requires vector quantities for position, displacement, velocity, and acceleration.

• Position Vector:

• Displacement:

• Magnitude of Displacement:

• Average Velocity:

• Instantaneous Velocity: Always tangent to the path;

• Speed:

• Average Acceleration:

Example: Displacement and Average Velocity

• An ant walks from to :

  • Magnitude:

  • If this takes ,

Relative Motion and Reference Frames

Motion is always measured relative to a chosen reference frame.

• Reference Frame: The environment in which measurements are made.

• Relative Velocity:

• Example: If your velocity relative to Earth is (east), Earth's velocity relative to you is (west).

Example: Boat Crossing a River

• A boat can travel at in still water. The river flows south at . To travel directly east, the boat must head at an angle north of east.

• Use vector addition to find the required heading and resultant speed.

• north of east.

• Resultant speed: eastward.

Two-Dimensional Motion with Constant Velocity and Acceleration

• Constant Velocity: ,

• Constant Acceleration: ,

• Velocity Components: ,

Example: Plane Takeoff

• A plane takes off at at . The horizontal speed (shadow's speed) is .

Example: Ball in a Computer Game

• Horizontal acceleration: ; vertical acceleration: ; initial position at origin; after :

Summary Table: Vector Operations

Key Takeaways

• Vectors are essential for describing motion in physics.

• Vector addition, subtraction, and decomposition are foundational skills.

• Two-dimensional kinematics involves analyzing motion using vector components and equations of motion.

• Relative motion problems require careful attention to reference frames and vector addition.