2.1 Scalars and Vectors
4 min read•june 24, 2024
Physics uses two types of quantities: scalars and . Scalars have only , like temperature or time. Vectors have both and direction, like velocity or . Understanding these is crucial for describing physical phenomena accurately.
Vectors can be added, subtracted, and multiplied by scalars. These operations are essential for solving physics problems involving multiple forces or displacements. algebra helps us analyze complex systems and predict outcomes in mechanics and other areas of physics.
Introduction to Scalars and Vectors
Scalars vs vectors in physics
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- Scalars quantify physical properties with a single value (magnitude) independent of direction
- Temperature (℃), time (seconds), speed (m/s), volume (liters), energy (joules)
- Vectors quantify physical properties with both magnitude and direction
- Represented by an arrow, length indicates magnitude, orientation indicates direction
- Displacement (meters north), velocity (m/s east), (m/s² down), force (newtons up), (kg⋅m/s southwest)
Components of vectors
- Magnitude quantifies the size or extent of the vector quantity
- Represented by the length of the vector arrow
- Always a positive value
- Direction specifies the orientation of the vector quantity in space
- Represented by the direction the vector arrow points
- Described using cardinal directions (north, southwest) or angles relative to a reference axis (30° above the x-axis)
- breaks a vector into its components along different axes
- Useful for analyzing complex motions or forces in multiple dimensions
Vector Operations and Applications
Scalar multiplication of vectors
- Multiplying a vector by a changes its magnitude without affecting its direction
- Positive scalar multiplication lengthens the vector (2 × doubles the length)
- Negative scalar multiplication reverses the vector's direction and lengthens it (-1 × flips and maintains length)
- The resulting vector is to the original vector
- Scalar multiplication formula:
- is the resulting vector
- is the scalar
- is the original vector
Vector addition and subtraction
- combines vectors graphically or analytically
- Graphically, place the tail of one vector at the head of the other
- Draw the from the tail of the first to the head of the
- Analytically, add the corresponding components:
- finds the difference between vectors graphically or analytically
- Graphically, place the tails of the vectors together
- Draw the resultant vector from the head of the subtracted vector to the head of the other vector
- Analytically, subtract the corresponding components:
Vector algebra for physics problems
- is the net force acting on an object due to multiple individual forces
- Found by vector addition of all the individual forces:
- A book on a table experiences the downward force of gravity and the upward normal force from the table
- is the net change in position due to multiple individual displacements
- Found by vector addition of all the individual displacements:
- A hiker's path can be broken into displacements (legs) and added to find the overall displacement (distance and direction) from the starting point
Interpretation of vector equations
- Vector equations describe relationships between
- 's second law, , relates force (a vector) to mass (a scalar) and acceleration (a vector)
- The direction of the force and acceleration vectors must match
- Scalar equations describe relationships between scalar quantities
- The kinetic energy equation, , relates kinetic energy (a scalar) to mass (a scalar) and speed (a scalar)
- Speed is the scalar magnitude of the
- Interpreting these equations requires understanding the physical meaning of each term and how they relate to the system being analyzed
Advanced Vector Operations
- (scalar product) of two vectors results in a scalar quantity
- Used to calculate work done by a force or to find the angle between two vectors
- of two vectors results in a vector to both original vectors
- Used in calculating torque or angular momentum
- Vector fields represent vector quantities that vary with position in space
- Examples include electric fields, magnetic fields, and fluid flow fields
Key Terms to Review (53)
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.
Action-at-a-distance force: An action-at-a-distance force is a force exerted by an object on another object that is not in physical contact with it, acting over a distance through space. Examples include gravitational, electromagnetic, and nuclear forces.
Antiparallel: Antiparallel vectors are vectors that have the same magnitude but point in opposite directions. They are important in understanding vector subtraction and relative motion.
Associative: Associative property in vectors states that the grouping of vectors does not affect their sum. Mathematically, for any vectors $a$, $b$, and $c$, $(a + b) + c = a + (b + c)$.
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.
Collinear: Collinear refers to points that lie on the same straight line. When vectors are described as collinear, it means they have the same or opposite direction and can be represented as multiples of one another. Understanding collinearity is essential for analyzing relationships between scalars and vectors, especially in situations involving displacement, velocity, and forces.
Commutative: Commutative refers to the property of an operation where changing the order of the operands does not change the result. In vector addition, for example, $\vec{A} + \vec{B} = \vec{B} + \vec{A}$.
Component Form: The component form of a vector is a way of representing the vector by its individual components or coordinates along the x, y, and z axes. This representation allows for the mathematical manipulation and analysis of vectors in a more systematic and quantitative manner.
Cross product: The cross product is a binary operation on two vectors in three-dimensional space, resulting in another vector that is perpendicular to the plane containing the original vectors. It is denoted by $\mathbf{A} \times \mathbf{B}$ and has both magnitude and direction.
Cross Product: The cross product, also known as the vector product, is a binary operation on two vectors that results in a third vector that is perpendicular to both of the original vectors. It is a fundamental concept in vector algebra and has important applications in various fields of physics, such as electromagnetism and mechanics.
Difference of two vectors: The difference of two vectors is found by subtracting the corresponding components of the vectors. The resulting vector points from the tip of the second vector to the tip of the first vector.
Displacement vector: A displacement vector represents the change in position of an object and has both magnitude and direction. It points from the initial position to the final position of the object.
Displacement Vector: A displacement vector is a vector quantity that represents the change in position of an object. It describes the shortest distance between an initial and final position, including both the magnitude and direction of the movement.
Distributive: The distributive property is a fundamental algebraic principle that allows for the multiplication of a scalar with a vector sum. Mathematically, it states that $a(b + c) = ab + ac$.
Dot product: The dot product is a scalar quantity obtained by multiplying the magnitudes of two vectors and the cosine of the angle between them. It provides a measure of how much one vector extends in the direction of another.
Dot Product: The dot product, also known as the scalar product, is a mathematical operation that combines two vectors to produce a scalar quantity. It is a fundamental concept in vector algebra that has important applications in various areas of physics, including the study of scalars, vectors, and their interactions.
Force: Force is a vector quantity that represents the interaction between two objects, causing a change in the motion or shape of the objects. It is the fundamental concept that underlies many of the physical principles studied in college physics, including Newton's laws of motion, work, energy, and more.
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.
Kepler’s second law: Kepler's second law, also known as the law of equal areas, states that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. This implies that a planet moves faster when it is closer to the Sun and slower when it is farther from the Sun.
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.
Momentum: Momentum is a vector quantity that describes the motion of an object. It is defined as the product of an object's mass and its velocity, and it represents the object's quantity of motion. Momentum is a fundamental concept in physics that is closely related to other important topics such as forces, energy, and collisions.
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 vectors: Orthogonal vectors are vectors that are perpendicular to each other. Their dot product is zero, indicating no projection of one vector onto the other.
Parallel: Parallel refers to objects or quantities that are side-by-side, occurring or existing at the same time, or having the same direction, course, or tendency. It is a fundamental concept in physics, particularly in the context of scalars and vectors.
Parallel vectors: Parallel vectors are vectors that have the same or exactly opposite direction. They can be scaled versions of each other, meaning one vector is a scalar multiple of the other.
Parallelogram law: The parallelogram law states that if two vectors are represented as two adjacent sides of a parallelogram, then the resultant vector can be represented by the diagonal of that parallelogram. This law is crucial for understanding how vectors combine, emphasizing the significance of both magnitude and direction when solving problems involving vector addition.
Perpendicular: Perpendicular refers to the relationship between two lines, vectors, or planes that intersect at a right angle, forming a 90-degree angle between them. This geometric concept is fundamental in understanding the behavior and properties of vectors, which are essential in the study of physics.
Resultant Displacement: Resultant displacement is the net or combined displacement of an object when multiple displacements are acting upon it. It represents the overall change in an object's position from its initial to final location, taking into account the individual displacements that contribute to the final outcome.
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.
Resultant vector: A resultant vector is the single vector that has the same effect as a combination of two or more vectors. It is found by vector addition.
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 equation: A scalar equation involves only scalar quantities, which are physical quantities that have magnitude but no direction. Scalar equations are often used to represent relationships in physics where direction is not a factor.
Second: The second is the base unit of time in the International System of Units (SI). It is defined as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom. The second is a fundamental unit that is essential for measuring and understanding various physical phenomena across the field of physics.
Tail-to-head geometric construction: Tail-to-head geometric construction is a method for adding vectors where the tail of each successive vector is placed at the head of the previous one. The resultant vector is drawn from the tail of the first vector to the head of the last vector.
The parallelogram rule: The parallelogram rule is a geometric method for adding two vectors. It involves placing the vectors tail to tail and completing the parallelogram formed by these vectors to find their resultant.
Triangle Law: The triangle law is a fundamental principle in vector mathematics that states the relationship between the vector addition of two or more vectors. It describes how the sum of two or more vectors can be represented geometrically as the third side of a triangle, with the original vectors forming the other two sides.
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.
Vector equation: A vector equation is a mathematical expression that represents physical quantities with both magnitude and direction using vectors. It is used to describe various phenomena in mechanics, sound, oscillations, and waves.
Vector Field: A vector field is a function that assigns a vector to every point in a given space. It is a mathematical representation of a physical quantity, such as a force or a velocity, that has both magnitude and direction at each point in the space.
Vector quantities: Vector quantities have both magnitude and direction. Examples include displacement, velocity, and force.
Vector Subtraction: Vector subtraction is the process of finding the difference between two vectors by subtracting the corresponding components of the vectors. It is a fundamental operation in vector algebra that allows for the manipulation and analysis of vector quantities.
Vector sum: The vector sum is the result of adding two or more vectors together, taking into account both their magnitudes and directions. It can be found using graphical methods or by breaking each vector into components.
Vectors: Vectors are quantities that have both magnitude and direction, often represented graphically by arrows. Common examples include displacement, velocity, and force.
Velocity vector: A velocity vector is a vector that describes both the speed and direction of an object's motion. It has both magnitude (speed) and direction, making it a fundamental quantity in kinematics.
Velocity Vector: The velocity vector is a vector quantity that describes the rate of change of an object's position with respect to time. It has both magnitude (speed) and direction, and is a fundamental concept in the study of motion and kinematics.