Vectors: Concepts, Addition, and Applications in Physics

Vectors in Physics

Definition and Properties

Vectors are fundamental mathematical objects in physics that possess both magnitude and direction. They are used to represent quantities such as displacement, velocity, acceleration, and force.

• Magnitude: The size or length of the vector, often measured in units appropriate to the physical quantity (e.g., meters for displacement).

• Direction: The orientation of the vector in space, typically specified by an angle or with respect to coordinate axes (e.g., north, east).

• Representation: Vectors are commonly depicted as arrows, where the length indicates magnitude and the arrowhead indicates direction.

Example: A velocity vector of 30 m/s east has a magnitude of 30 m/s and points east.

Vector Addition

Head-to-Tail Method

Vectors can be added together to find a resultant vector. The most common graphical method is the head-to-tail method, where the tail of one vector is placed at the head of the previous vector.

• Procedure: Place the tail of the second vector at the head of the first. The resultant vector is drawn from the tail of the first to the head of the last.

• Resultant Vector: The single vector that has the same effect as the original vectors combined.

Example: If a car drives 90 km due east and then 200 km due east, the resultant displacement vector is 290 km due east.

Vector Addition Using Components

When vectors are not aligned along the same direction, they can be added using their components along the coordinate axes (usually x and y).

• Component Form: Any vector A can be written as $A_x$ and $A_y$, where $A_x$ is the component along the x-axis and $A_y$ along the y-axis.

• Equations:

  • $A_x = A \\cos\theta$

  • $A_y = A \\sin\theta$

• Resultant Vector: For vectors A and B:

  • $R_x = A_x + B_x$

  • $R_y = A_y + B_y$

  • $R = \\sqrt{R_x^2 + R_y^2}$

  • $\\theta = \\tan^{-1}(R_y / R_x)$

Example: A person walks 125 m north and then 75 m south. The resultant displacement is 50 m north.

Pythagorean Theorem and Trigonometry in Vector Addition

When vectors are perpendicular, the magnitude of the resultant can be found using the Pythagorean theorem.

• Equation: $R = \\sqrt{A^2 + B^2}$ (for vectors at 90°)

• Direction: $\\theta = \\tan^{-1}(B/A)$

Example: An airplane flies 300 km due east and then 700 km due north. The resultant displacement is:

• $R = \\sqrt{(300)^2 + (700)^2} = 762$ km

• Direction: $\\theta = \\tan^{-1}(700/300) = 66.8°$ north of east

Cardinal Directions and Angled Motion

Describing Directions

Vectors in physics are often described using the four cardinal directions: North (N), South (S), East (E), and West (W). For angled motion, headings such as northeast (NE) or 30° east of north are used.

• Special Designations: NE, SE, NW, SW indicate directions at 45° between cardinal points.

• Angle Descriptions: Angles are described by stating the closest cardinal direction and the direction towards which the angle is measured (e.g., "30° east of north").

Example: A vector at 30° east of north points 30° towards east from the north direction.

Vector Components and Trigonometric Ratios

Resolving Vectors into Components

Any vector can be resolved into perpendicular components using trigonometric ratios:

• Sine: $\\sin\\theta = \\frac{\\text{Opposite}}{\\text{Hypotenuse}}$

• Cosine: $\\cos\\theta = \\frac{\\text{Adjacent}}{\\text{Hypotenuse}}$

• Tangent: $\\tan\\theta = \\frac{\\text{Opposite}}{\\text{Adjacent}}$

Application: Used to find the x and y components of a vector given its magnitude and direction.

Free-Body Diagrams and Vector Applications

Forces as Vectors

In physics, forces are represented as vectors in free-body diagrams. Each force has a magnitude and direction, and the net force is found by vector addition.

• Example: A ring is pulled by two ropes at different angles. The tension forces $T_1$ and $T_2$ are resolved into components and added to find the net force.

• Component Equations:

  • $T_1$ at 10°: $T_1 \\cos 10°$ (vertical), $T_1 \\sin 10°$ (horizontal)

  • $T_2$ at 80°: $T_2 \\cos 80°$ (vertical), $T_2 \\sin 80°$ (horizontal)

Worked Example: Displacement with Angled Vectors

Calculating Resultant Displacement

Suppose a person runs 145 m in a direction 20.0° north of east (vector A) and then 215 m in a direction 35.0° south of east (vector B). To find the magnitude and direction of the resultant vector C:

• Resolve each vector into x and y components:

  • For vector A:

    • $A_x = 145 \\cos 20.0°$

    • $A_y = 145 \\sin 20.0°$

  • For vector B:

    • $B_x = 215 \\cos 35.0°$

    • $B_y = -215 \\sin 35.0°$ (negative because south of east)

• Add components to get resultant:

  • $C_x = A_x + B_x$

  • $C_y = A_y + B_y$

• Find magnitude and direction:

  • $C = \\sqrt{C_x^2 + C_y^2}$

  • $\\theta = \\tan^{-1}(C_y / C_x)$

Summary Table: Vector Operations

Operation	Method	Equation
Addition (same direction)	Algebraic sum	$R = A + B$
Addition (perpendicular)	Pythagorean theorem	$R = \\sqrt{A^2 + B^2}$
Component addition	Sum x and y components	$R_x = A_x + B_x$, $R_y = A_y + B_y$
Magnitude from components	Pythagorean theorem	$R = \\sqrt{R_x^2 + R_y^2}$
Direction from components	Trigonometry	$\\theta = \\tan^{-1}(R_y / R_x)$

Additional info: Some context and equations have been inferred and expanded for completeness and clarity.