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2.2 Coordinate Systems and Components of a Vector

2 min read•june 24, 2024

Vectors are crucial in physics, allowing us to represent quantities with both and . They're used to describe forces, velocities, and more. Understanding how to break down vectors into components and work with different coordinate systems is key.

Coordinate systems help us describe quantities in space. Whether using Cartesian (x, y, z) or , these systems let us analyze complex motions and interactions. Mastering vector components and coordinate transformations is essential for solving physics problems.

Coordinate Systems and Vector Components

Vectors in multiple dimensions

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  • Vectors in two dimensions represented by components along x and y axes in a
    • v⃗=vxi^+vyj^\vec{v} = v_x \hat{i} + v_y \hat{j} where vxv_x and vyv_y are components and i^\hat{i} and j^\hat{j} are unit vectors along x and y axes (, force)
  • Vectors in three dimensions represented by components along x, y, and z axes
    • v⃗=vxi^+vyj^+vzk^\vec{v} = v_x \hat{i} + v_y \hat{j} + v_z \hat{k} where vxv_x, vyv_y, and vzv_z are components and i^\hat{i}, j^\hat{j}, and k^\hat{k} are unit vectors along x, y, and z axes (, )
  • These coordinate axes are , meaning they are perpendicular to each other

Vector vs scalar components

  • Vector components are directed quantities with both and direction
    • Represented by vxi^v_x \hat{i}, vyj^v_y \hat{j}, and vzk^v_z \hat{k} (velocity components, force components)
  • Scalar components are numerical values without direction
    • Represented by vxv_x, vyv_y, and vzv_z (, )

Magnitude calculation from components

  • Two-dimensional vectors magnitude calculated using
    • ∣v⃗∣=vx2+vy2|\vec{v}| = \sqrt{v_x^2 + v_y^2} (, )
  • Three-dimensional vectors magnitude calculated using extension of Pythagorean theorem
    • ∣v⃗∣=vx2+vy2+vz2|\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2} (resultant acceleration, )

Direction angles of planar vectors

  • θ\theta measured counterclockwise from positive x-axis
  • Calculated using tangent function
    1. tan⁡θ=vyvx\tan \theta = \frac{v_y}{v_x}
    2. θ=tan⁡−1(vyvx)\theta = \tan^{-1} (\frac{v_y}{v_x})
  • Quadrant adjustments may be necessary based on signs of vxv_x and vyv_y (force direction, velocity direction)

Polar vs Cartesian coordinates

  • (r,θ)(r, \theta) specify magnitude rr and direction angle θ\theta measured counterclockwise from positive x-axis (, )
  • (x,y)(x, y) specify horizontal component xx and vertical component yy (, )
  • Converting from polar to Cartesian coordinates
    • x=rcos⁡θx = r \cos \theta
    • y=rsin⁡θy = r \sin \theta
  • Converting from Cartesian to polar coordinates
    • r=x2+y2r = \sqrt{x^2 + y^2}
    • θ=tan⁡−1(yx)\theta = \tan^{-1} (\frac{y}{x}) with quadrant adjustments

Vector Analysis Techniques

  • : Breaking down vectors into their x and y components for easier calculations
  • : Separating a vector into its component parts along chosen axes
  • is essential for relating vector components to the original vector's magnitude and direction

Key Terms to Review (52)

Acceleration: Acceleration is the rate of change of velocity with respect to time. It represents the change in an object's speed or direction over a given time interval, and is a vector quantity that has both magnitude and direction.
Acceleration vector: An acceleration vector is a vector quantity that represents the rate of change of velocity of an object. It indicates both the magnitude and direction of this change.
Average speed: Average speed is the total distance traveled divided by the total time taken to travel that distance. It is a scalar quantity and does not take direction into account.
Cartesian Coordinates: Cartesian coordinates are a system used to locate points in space by specifying their positions along orthogonal (perpendicular) axes. This coordinate system provides a mathematical framework for representing and analyzing vectors, which are essential in understanding the behavior of physical quantities such as displacement, velocity, and acceleration.
Circular Motion: Circular motion is the movement of an object in a circular path or trajectory, where the object continuously changes direction while maintaining a constant distance from the center of the circle. This type of motion is characterized by a centripetal force that acts on the object, causing it to follow a curved path rather than a straight line.
Component Method: The component method is a technique used to break down and analyze vectors by separating them into their individual components along specific coordinate axes. This method allows for the study of vector quantities in the context of coordinate systems and their corresponding vector components.
Coordinate plane: A coordinate plane is a two-dimensional surface where points are defined by pairs of numerical values, known as coordinates, typically represented as (x, y). This system allows for the visualization and representation of mathematical concepts, including vectors, which can be broken down into their components based on their position in relation to the axes of the plane.
Coordinate system: A coordinate system is a framework for determining the position of points or vectors in space using numbers, typically defined by axes. Common types include Cartesian and polar coordinate systems.
Direction: Direction refers to the orientation of a vector in space, indicating where it is pointing relative to a reference point or coordinate system. It is crucial in understanding how vectors represent physical quantities like displacement, velocity, and acceleration, as each of these requires both magnitude and direction for a complete description.
Direction angle: A direction angle is the angle between a reference axis, typically the positive x-axis, and a vector in a coordinate system. It is used to describe the orientation of the vector in space.
Displacement: Displacement is a vector quantity that refers to the change in position of an object. It is measured as the straight-line distance from the initial to the final position, along with the direction.
Displacement: Displacement is the change in position of an object relative to a reference point. It is a vector quantity, meaning it has both magnitude and direction, and is used to describe the movement of an object in physics.
Electric Field Strength: Electric field strength is a vector quantity that describes the magnitude and direction of the electric force acting on a unit positive charge at a given point in an electric field. It is a fundamental concept in electromagnetism that is crucial for understanding the behavior of charged particles and the interactions between electric fields and matter.
Equilibrium: Equilibrium occurs when all forces acting on an object are balanced, resulting in no net force and no acceleration. In static equilibrium, the object is at rest, and in dynamic equilibrium, it moves with constant velocity.
Equilibrium: Equilibrium is a state of balance or stability, where the forces acting on a system are in balance, and the system is at rest or in a state of constant motion. This concept is fundamental in understanding various physical phenomena, including the behavior of objects, the distribution of forces, and the stability of systems.
Free-body diagram: A free-body diagram is a graphical representation used to visualize the forces acting on an object. Each force is represented by an arrow pointing in the direction of the force with its length proportional to the magnitude.
Free-Body Diagram: A free-body diagram is a visual representation of an object or system that shows all the external forces acting on it. It is a fundamental tool used in physics to analyze the forces acting on an object and to solve problems involving Newton's laws of motion.
Inclined plane: An inclined plane is a flat surface tilted at an angle to the horizontal. It is used to facilitate raising or lowering a load with less effort.
Inclined Plane: An inclined plane is a flat surface that is tilted or angled relative to the horizontal. It is a simple machine that is used to lift or move objects by applying a force parallel to the surface, rather than perpendicular to it. The inclined plane is a fundamental concept in physics, with applications across various topics.
Joule: A joule is the SI unit of work or energy, equivalent to one newton-meter. It represents the amount of work done when a force of one newton displaces an object by one meter in the direction of the force.
Joule: The joule (J) is the standard unit of energy in the International System of Units (SI). It represents the amount of work done or energy expended when a force of one newton acts through a distance of one meter.
Magnetic Field: A magnetic field is a region around a magnetic material or a moving electric charge within which the force of magnetism acts. It can be visualized as lines that emanate from magnetic poles, influencing the behavior of charged particles and magnetic materials within that region. The direction and strength of a magnetic field can be represented using vectors, making it important to understand its components in different coordinate systems.
Magnitude: Magnitude is the size or length of a vector, representing its absolute value. It is always a non-negative scalar quantity.
Magnitude: Magnitude is a quantitative measure that describes the size, scale, or extent of a physical quantity. It is a fundamental concept in physics that is essential for understanding and analyzing various physical phenomena.
Mass: Mass is a fundamental physical quantity that represents the amount of matter in an object. It is a measure of an object's resistance to changes in its state of motion, and it is a key concept in the study of mechanics and the behavior of objects under the influence of forces.
Newton: Newton is the standard unit of force in the International System of Units (SI), named after the renowned English physicist and mathematician, Sir Isaac Newton. It is a fundamental unit that is essential in understanding and describing the behavior of objects under the influence of various forces, as well as in the study of mechanics, dynamics, and other related areas of physics.
Orthogonal: Orthogonal refers to the property of being at right angles or perpendicular to one another. This term is particularly important in the context of coordinate systems and the components of a vector, as it describes the relationship between the axes or dimensions that define a coordinate system or the components that make up a vector.
Pendulum Motion: Pendulum motion refers to the oscillatory movement of a weight suspended from a fixed point by a string or rod. This periodic motion is governed by the principles of classical mechanics and is a fundamental concept in the study of physics.
Polar coordinate system: The polar coordinate system represents points in a plane using a distance from a reference point (the origin) and an angle from a reference direction (usually the positive x-axis). It is commonly used in physics to simplify the analysis of rotational systems and wave functions.
Polar coordinates: Polar coordinates represent a point in a plane using two values: the radial distance from a reference point (origin) and the angle from a reference direction. They are useful for problems involving circular or rotational symmetry.
Polar Coordinates: Polar coordinates are a two-dimensional coordinate system that uses a distance from a fixed point, called the pole, and an angle to specify the location of a point. This coordinate system is an alternative to the more commonly used Cartesian coordinate system, which uses perpendicular x and y axes.
Projectile motion: Projectile motion is the motion of an object thrown or projected into the air, subject to only the acceleration due to gravity. It involves two components of motion: horizontal and vertical.
Projectile Motion: Projectile motion is the motion of an object that is launched into the air and moves solely under the influence of gravity and without any additional force acting on it. It is a type of motion that follows a curved trajectory, with the object's position and velocity changing over time in a predictable manner.
Pythagorean theorem: The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. This theorem is fundamental in connecting geometry to physics, especially when dealing with problems involving distances and angles.
Radial coordinate: A radial coordinate is a measure of distance from a fixed origin point in a polar or spherical coordinate system. It specifies the length along the radius vector from the origin to a given point.
Resultant Force: The resultant force is the single force that represents the combined effect of all individual forces acting on an object. This concept is crucial for understanding how forces interact and affect motion, as it simplifies complex situations involving multiple forces into a single equivalent force, which can be analyzed further. It plays a key role in connecting the ideas of scalars and vectors, as well as how forces can be broken down into components based on coordinate systems.
Scalar: A scalar is a quantity that has only magnitude and no direction. Common examples include mass, temperature, and time.
Scalar: A scalar is a physical quantity that has only magnitude and no direction. It is a single numerical value that represents the size or amount of something, without any associated directional information.
Scalar component: A scalar component is the projection of a vector onto an axis, represented as a single numerical value. It quantifies how much of the vector lies along that specific axis.
Speed: Speed is a scalar quantity that describes the rate of change of an object's position with respect to time. It is a fundamental concept in physics that is essential for understanding motion and the behavior of objects in various contexts.
The component form of a vector: The component form of a vector expresses the vector in terms of its horizontal and vertical components. This representation is useful for performing vector addition, subtraction, and other operations.
Trigonometry: Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental tool in the study of physics, particularly in the understanding of vectors and their components within coordinate systems.
Uniform circular motion: Uniform circular motion occurs when an object moves in a circular path with constant speed. While the speed is constant, the direction of the object's velocity continuously changes.
Unit Vector: A unit vector is a dimensionless vector with a magnitude of 1 that points in a specific direction. It is used to represent the direction of a vector without regard to its magnitude.
Unit vectors of the axes: Unit vectors of the axes are vectors that have a magnitude of 1 and point in the direction of the coordinate axes. They are typically denoted as $\hat{i}$, $\hat{j}$, and $\hat{k}$ in three-dimensional space for the x, y, and z-axes respectively.
Vector: A vector is a mathematical quantity that has both magnitude (size or length) and direction. It is used to represent physical quantities in physics, such as displacement, velocity, acceleration, and force, where both the size and the direction of the quantity are important.
Vector Addition: Vector addition is the process of combining two or more vectors to obtain a single vector that represents their combined effect. This fundamental concept is essential in understanding the behavior of physical quantities that have both magnitude and direction, such as displacement, velocity, and acceleration.
Vector Decomposition: Vector decomposition is the process of breaking down a vector into its component parts along specific coordinate axes or reference frames. This technique is essential for analyzing and manipulating vectors in various fields, including physics, engineering, and mathematics.
Velocity: Velocity is a vector quantity that describes the rate of change of an object's position with respect to time. It includes both the speed and the direction of an object's motion, making it a more complete description of an object's movement compared to just speed alone.
X-Component: The x-component of a vector is the projection of the vector onto the x-axis of a coordinate system. It represents the horizontal or left-right component of the vector's magnitude and direction.
Y-component: The y-component of a vector represents the vertical or up-down direction of the vector. It is one of the two or three components that describe the magnitude and direction of a vector in a coordinate system.
Z-component: The z-component of a vector is the scalar projection of the vector onto the z-axis of a coordinate system. It represents the magnitude or length of the vector's projection along the vertical or depth dimension.
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2.3 Algebra of Vectors