2.2 Vectors, Scalars, and Coordinate Systems
2 min read•june 18, 2024
Vectors and scalars are fundamental concepts in physics. Vectors have both and direction, like velocity, while scalars only have magnitude, like . Understanding these helps us describe motion and forces accurately.
Coordinate systems provide a framework for measuring position and motion. One-dimensional systems use a single axis, with positive and negative directions. This allows us to track an object's position, displacement, and direction of movement.
Vectors and Scalars
Scalar vs vector quantities
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- quantities characterized by magnitude only no direction (, , )
- quantities have both magnitude and direction (displacement, velocity, acceleration)
- examples include speed which is the magnitude of velocity, a measure of an object's capacity to do work
- examples include which has both magnitude and direction, the product of mass and velocity
Coordinate systems for motion
- occurs along a straight line described using a single coordinate axis
- Positive direction typically chosen as right or upward, objects moving this way have positive displacement, velocity, acceleration
- Negative direction is opposite, objects moving this way have negative displacement, velocity, acceleration
- is the reference point where position is zero
- Position denoted by variable is location of object relative to origin
- Displacement is change in position, positive when moving in positive direction, negative in negative direction
Vector notation in physics
- Vectors represented with arrow above symbol ( for velocity, for acceleration, for force)
- Magnitude is absolute value of vector ( for speed, for acceleration magnitude, for force magnitude)
- Direction specified by angle or relative to (, )
- Vectors can be decomposed into components along coordinate axes ()
- and are unit vectors in and directions
- Vector addition involves combining multiple vectors to determine a resultant vector
Vector components and coordinate systems
- use perpendicular axes (x, y, z) to describe
- use magnitude and angle to describe vectors, useful for circular motion
- Vector components are projections of a vector onto coordinate axes
- Magnitude of a vector can be calculated from its components using the Pythagorean theorem
Coordinate Systems
Coordinate systems for motion
- One-dimensional motion occurs along straight line, described with single coordinate axis
- Positive direction usually right or upward
- Negative direction is opposite
- Origin is reference point where position is zero
- Position is object's location relative to origin
- Displacement is change in position
- Positive when moving in positive direction
- Negative when moving in negative direction
Key Terms to Review (37)
$ ext{Delta } x$: $ ext{Delta } x$ is the change in position or displacement along the $ ext{x}$-axis in a coordinate system. It represents the difference between an object's final and initial positions in the $ ext{x}$-direction, and is a fundamental concept in the study of vectors, scalars, and coordinate systems.
$ ext{hat{i}}$: $ ext{hat{i}}$ is a unit vector in the positive x-direction of a Cartesian coordinate system. It is a vector of length 1 that points along the x-axis, and is used to represent the direction of a vector in two- or three-dimensional space.
$ ext{hat}{j}$: $ ext{hat}{j}$ is a unit vector in the positive y-direction of a Cartesian coordinate system. It represents the direction of the y-axis and is used to describe the orientation and magnitude of vectors in two-dimensional space.
The term $ ext{hat}{j}$ is an essential component in the understanding of vectors, scalars, and coordinate systems, as it provides a standardized way to represent and manipulate vector quantities.
$ ext{vec}(F)$: $ ext{vec}(F)$ is a vector quantity that represents force, which is the interaction between two objects that can cause the objects to change their velocity, shape, or direction of motion. It is a fundamental concept in physics that describes the push or pull experienced by an object due to the action of another object.
$ ext{vec}(v)$: $ ext{vec}(v)$ is a vector quantity that represents the velocity or rate of change in position of an object over time. It has both magnitude, which is the speed or rate of change, and direction, which indicates the path the object is moving along.
$\vec{a}$: $\vec{a}$ represents acceleration, a vector quantity that indicates the rate of change of velocity of an object over time. It not only has a magnitude, which describes how fast the velocity changes, but also a direction, which shows the direction in which the object's velocity is changing. Understanding acceleration is crucial in physics, as it ties together concepts like force, mass, and motion.
$|\vec{a}|$: $|\vec{a}|$ represents the magnitude or absolute value of a vector $\vec{a}$. It is a scalar quantity that describes the length or size of the vector, without regard to its direction. The magnitude of a vector is a fundamental concept in the study of vectors, scalars, and coordinate systems.
$|\vec{F}|$: $|\vec{F}|$ represents the magnitude or absolute value of a vector quantity $\vec{F}$. It is a scalar value that describes the size or strength of the vector, without regard to its direction. This term is essential in understanding the concept of vectors and their applications in physics.
$|\vec{v}|$: $|\vec{v}|$ represents the magnitude or absolute value of a vector $\vec{v}$. It is a scalar quantity that gives the length or size of the vector, without any directional information. The magnitude of a vector is a fundamental concept in the study of vectors, scalars, and coordinate systems.
$F_x$: $F_x$ represents the component of a force vector acting along the x-axis in a Cartesian coordinate system. This term is essential for understanding how forces can be broken down into their respective components, allowing for easier calculations and analysis of motion in one or multiple dimensions. By isolating the x-component of a force, we can apply Newton's laws of motion more effectively to solve problems involving forces acting at angles or in different directions.
$F_y$: $F_y$ represents the vertical component of a force vector acting in the y-direction within a coordinate system. In physics, forces are often expressed in terms of their components along the x and y axes, allowing for easier analysis of motion and equilibrium. Understanding $F_y$ is crucial for breaking down complex forces into simpler, manageable parts and applying Newton's laws effectively.
Adhesive forces: Adhesive forces are the attractive forces between unlike molecules. They play a significant role in phenomena such as capillary action and the wetting of surfaces.
Approximations: Approximations are simplified representations or calculations that are close to the exact value, used when exact values are impractical. They help in making complex problems more manageable.
Atomic mass: Atomic mass is the weighted average mass of an atom's naturally occurring isotopes, measured in atomic mass units (amu). It reflects both the mass and relative abundance of each isotope.
Cartesian Coordinates: Cartesian coordinates are a system used to locate points in a two-dimensional or three-dimensional space. This system uses perpendicular axes, known as the x-axis and y-axis in two dimensions, or the x-axis, y-axis, and z-axis in three dimensions, to uniquely identify the position of a point.
Coordinate System: A coordinate system is a mathematical framework used to uniquely identify the position of a point or an object in space. It provides a systematic way to describe the location of an entity relative to a defined origin and a set of reference axes.
Critical temperature: Critical temperature is the highest temperature at which a substance can exist as a liquid, regardless of pressure. Beyond this temperature, the substance becomes a supercritical fluid.
Displacement Vector: A displacement vector is a directed line segment that represents the change in position of an object. It describes the distance and direction an object has moved from its initial location to its final location.
Energy: Energy is the capacity to do work or cause change. It is the fundamental currency that powers all physical and chemical processes in the universe, from the motion of subatomic particles to the dynamics of entire galaxies. Energy is a unifying concept that connects diverse areas of physics, including mechanics, thermodynamics, electromagnetism, and quantum mechanics.
Force: Force is a vector quantity that represents the interaction between two objects, causing a change in the motion or shape of one or both objects. It is a fundamental concept in physics that describes the push or pull experienced by an object due to the influence of another object or system.
Magnitude: Magnitude is a measure of the size or scale of a quantity, representing its absolute or relative value. It is a fundamental concept in physics, particularly in the context of vectors, forces, and displacement.
Mass: Mass is a fundamental property of matter that represents the amount of material in an object. It is a measure of the object's resistance to changes in its motion, and it is a scalar quantity, meaning it has magnitude but no direction. Mass is a crucial concept in physics, as it is a key factor in determining an object's behavior under the influence of forces.
Momentum: Momentum is a vector quantity that represents the product of an object's mass and velocity. It is a measure of an object's quantity of motion and is conserved in a closed system, meaning the total momentum of a system remains constant unless acted upon by an external force.
One-Dimensional Motion: One-dimensional motion refers to the movement of an object along a single axis, typically the x-axis, without any changes in the other spatial dimensions (y and z). This type of motion is often used as a simplified model to understand the basic principles of kinematics and dynamics, which are the study of motion and the forces that cause it.
Origin: The origin is the point of reference from which measurements and calculations are made in a coordinate system. It serves as the starting point for defining the position and orientation of objects within a given frame of reference.
Polar Coordinates: Polar coordinates are a two-dimensional coordinate system that uses the distance from a reference point, called the pole, and the angle from a reference direction, called the polar axis, to locate a point on a plane. This system provides an alternative to the more commonly used Cartesian coordinate system.
Position Vector: A position vector is a vector that represents the location of a point in space relative to a chosen origin or reference point. It is a fundamental concept in the study of vectors, coordinate systems, and the description of motion in physics.
Scalar: A scalar is a physical quantity that has only magnitude and no direction. Examples include mass, temperature, and electric potential.
Scalar: A scalar is a physical quantity that has only a magnitude, or numerical value, and no direction. Scalars are contrasted with vectors, which have both a magnitude and a direction. Scalars are commonly used in physics to describe various physical properties and quantities.
Speed: Speed is a scalar quantity that measures how fast an object moves, defined as the distance traveled per unit of time. It represents the rate at which an object covers distance and does not include any information about the direction of the movement. Understanding speed is crucial in analyzing motion, as it relates to both vectors and coordinate systems by helping to describe an object's path and position over time.
Temperature: Temperature is a physical quantity that measures the average kinetic energy of the particles, such as atoms or molecules, in a substance. It is a fundamental concept that is closely related to the behavior of matter and energy in various contexts, including vectors, scalars, coordinate systems, the ideal gas law, kinetic theory, and phase changes.
Time: Time is a fundamental concept in physics that describes the progression of events and the duration of phenomena. It is a measure of the interval between two events or the length of an occurrence, and it is essential in understanding and analyzing various physical processes, including motion, acceleration, and the behavior of systems over time.
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 considering its magnitude.
Vector: A vector is a quantity that has both magnitude and direction. It is typically represented by an arrow where the length represents the magnitude and the arrowhead points in the direction.
Vector: A vector is a mathematical quantity that has both magnitude (size) and direction. Vectors are used to describe physical quantities in physics, such as displacement, velocity, and force, where both the size and direction of the quantity are important.
Vector Components: Vector components are the individual parts or projections of a vector along the coordinate axes. They represent the magnitudes of a vector in each of the coordinate directions, allowing for the vector to be broken down and analyzed in terms of its individual directional quantities.
Velocity vector: A velocity vector is a quantity that represents both the speed and direction of an object's motion. It combines the scalar magnitude, which indicates how fast the object is moving, with a directional component that specifies the direction in which the object is traveling. This dual nature makes the velocity vector essential in understanding motion and analyzing how different velocities can interact with one another.