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Vectors are the superheroes of mathematics, swooping in to tackle problems involving and . They're not just lines with arrows; they're powerful tools for representing everything from forces in physics to data in computer graphics.

Understanding vectors is like unlocking a secret language of motion and space. We'll explore how to add, multiply, and manipulate these mathematical marvels, seeing how they simplify complex problems and bring abstract concepts to life in the real world.

Vector Fundamentals

Vectors as geometric objects

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  • Represented as directed line segments
    • Length represents 's magnitude (size or length)
    • Arrow indicates vector's direction (orientation in space)
  • Denoted using boldface letters (v\mathbf{v}) or letters with arrows (v⃗\vec{v})
  • Magnitude of vector v⃗\vec{v} denoted as ∣v⃗∣|\vec{v}| (absolute value or norm)
  • Two vectors equal if same magnitude and direction (regardless of position)

Vector magnitude and direction

  • Magnitude of vector v⃗=(x,y)\vec{v} = (x, y) in plane given by formula:
    • ∣v⃗∣=x2+y2|\vec{v}| = \sqrt{x^2 + y^2} (Pythagorean theorem)
  • Direction described using:
    • Angle measures (45° counterclockwise from positive x-axis)
    • Cardinal directions (northeast, southwest)
  • Direction of vector v⃗=(x,y)\vec{v} = (x, y) found using arctangent function:
    • θ=tan⁡−1(yx)\theta = \tan^{-1}(\frac{y}{x}), where θ\theta is angle with positive x-axis
    • Determines quadrant vector points to (I, II, III, or IV)

Basic vector operations

  • Vector addition: To add vectors u⃗=(u1,u2)\vec{u} = (u_1, u_2) and v⃗=(v1,v2)\vec{v} = (v_1, v_2), add corresponding components:
    • u⃗+v⃗=(u1+v1,u2+v2)\vec{u} + \vec{v} = (u_1 + v_1, u_2 + v_2) (component-wise addition)
    • Visualized using parallelogram rule or triangle rule (head-to-tail method)
  • : To multiply vector v⃗=(x,y)\vec{v} = (x, y) by scalar cc, multiply each component by cc:
    • cv⃗=(cx,cy)c\vec{v} = (cx, cy) (distributes to each component)
    • Changes magnitude but not direction (unless cc negative)

Vector Representation and Products

Vectors in component form

  • Standard unit vectors in plane:
    • i^=(1,0)\hat{i} = (1, 0), points in positive x-direction (horizontal)
    • j^=(0,1)\hat{j} = (0, 1), points in positive y-direction (vertical)
  • Vector v⃗=(x,y)\vec{v} = (x, y) expressed in as:
    • v⃗=xi^+yj^\vec{v} = x\hat{i} + y\hat{j} ( of unit vectors)
  • Allows for easy vector addition and (algebraic operations)

Dot product of vectors

  • of vectors u⃗=(u1,u2)\vec{u} = (u_1, u_2) and v⃗=(v1,v2)\vec{v} = (v_1, v_2) defined as:
    • u⃗⋅v⃗=u1v1+u2v2\vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 (sum of products of corresponding components)
  • Results in scalar value (not a vector)
  • Related to angle θ\theta between vectors:
    • u⃗⋅v⃗=∣u⃗∣∣v⃗∣cos⁡(θ)\vec{u} \cdot \vec{v} = |\vec{u}||\vec{v}|\cos(\theta) (magnitude and direction)
  • Used to determine:
    • Orthogonality (perpendicularity): u⃗⋅v⃗=0\vec{u} \cdot \vec{v} = 0 (90° angle)
    • Projection of one vector onto another (component in direction of other vector)

Unit vectors in direction

  • has magnitude of 1 (length of 1)
  • To construct unit vector u^\hat{u} in direction of vector v⃗=(x,y)\vec{v} = (x, y):
    • Divide v⃗\vec{v} by its magnitude: u^=v⃗∣v⃗∣=(xx2+y2,yx2+y2)\hat{u} = \frac{\vec{v}}{|\vec{v}|} = (\frac{x}{\sqrt{x^2 + y^2}}, \frac{y}{\sqrt{x^2 + y^2}}) (normalization)
  • Useful for representing directions without considering magnitude (pure direction)

Applications of Vectors

Vector applications in physics

  • Represent physical quantities with magnitude and direction:
    • Displacement, velocity, acceleration (kinematics)
    • Force, momentum (dynamics)
  • Analyze and solve problems involving:
    1. Find resultant force acting on object
    2. Express each in component form
    3. Add force vectors component-wise to find resultant force vector
    4. Calculate magnitude and direction of resultant force vector
  • Motion in 2D or 3D (projectile motion, circular motion)
  • Work done by force along displacement (dot product)
  • Vector fields describe physical quantities that vary in space (e.g., electromagnetic fields)

Advanced Vector Concepts

Vector spaces and calculus

  • : A set of vectors that can be added and scaled, following specific axioms
  • Linear combination: Expressing a vector as a sum of scaled vectors from a given set
  • : Branch of mathematics dealing with differentiation and integration of vector fields
  • Coordinate systems: Frameworks for specifying vector positions (e.g., Cartesian, polar, spherical)

Key Terms to Review (29)

Basis Vectors: Basis vectors are a set of linearly independent vectors that form the foundation for representing any vector in a given vector space. They provide a coordinate system that allows vectors to be expressed as a unique linear combination of these basis vectors.
Component Form: The component form of a vector is a way to represent the vector using its individual components or coordinates. This representation allows for the mathematical manipulation and analysis of vectors in various contexts, such as in the study of vectors in 10.8 Vectors.
Coordinate System: A coordinate system is a mathematical framework used to represent and locate points or objects in space. It provides a systematic way to assign unique coordinates, such as (x,y) or (x,y,z), to every point within a defined region or space.
Cross Product: The cross product, also known as the vector product, is a binary operation in three-dimensional Euclidean space that takes two vectors and produces a third vector that is perpendicular to both of the original vectors. The cross product is an important concept in the study of vectors and their applications in physics and mathematics.
Curl: Curl is a vector calculus operation that describes the circulation or rotation of a vector field around a given point. It is a measure of the tendency of the vector field to curl or spin around that point.
Direction: Direction refers to the path or orientation of a vector, indicating the way in which a quantity, such as a force or velocity, is moving or pointing. It is a fundamental concept in the study of vectors, which are mathematical representations of quantities with both magnitude and direction.
Divergence: Divergence is a mathematical concept that describes the rate at which a vector field is expanding or contracting at a given point. It measures the density of the outward flux of a vector field from an infinitesimal volume around a given point. Divergence is an important concept in the study of vector calculus, fluid dynamics, and electromagnetism.
Dot Product: The dot product, also known as the scalar product, is a mathematical operation performed on two vectors that results in a scalar (single numerical) value. It is a fundamental concept in the study of vectors, which are essential in various fields, including physics, engineering, and computer science.
Force Vector: A force vector is a mathematical representation of a force that includes both the magnitude (size) and direction of the force acting on an object. It is a fundamental concept in the study of physics and engineering, particularly in the context of analyzing and describing the motion of objects under the influence of various forces.
Gibbs: Gibbs is a concept in physics and mathematics that describes the behavior of vector fields, particularly in the context of electromagnetic theory and fluid dynamics. It provides a framework for understanding and analyzing the properties of vector fields, such as their divergence, curl, and potential functions.
Gradient: The gradient of a vector field is a vector that points in the direction of the greatest rate of increase of the field, and whose magnitude is the rate of change in that direction. It is a fundamental concept in vector calculus and has applications in various areas of mathematics and physics.
Hamilton: Hamilton is a mathematical concept that refers to the vector product, also known as the cross product, of two vectors. It is a fundamental operation in the study of vectors and has important applications in various fields, including physics, engineering, and mathematics.
I-j-k Notation: The i-j-k notation, also known as the standard basis or unit vector notation, is a way of representing and working with vectors in a three-dimensional coordinate system. It provides a standardized and intuitive framework for describing the direction and magnitude of vectors using three perpendicular unit vectors: i, j, and k.
Linear Combination: A linear combination is the sum of a set of vectors, each multiplied by a corresponding scalar (numerical) coefficient. It represents a way of combining multiple vectors into a single vector by applying specific weights or coefficients to each vector.
Linear Independence: Linear independence is a fundamental concept in linear algebra that describes a set of vectors or functions that are not related by any linear combination. In other words, no vector or function in the set can be expressed as a linear combination of the others.
Magnitude: Magnitude is a quantitative measure that describes the size or scale of a vector. It represents the length or absolute value of a vector, indicating its strength or intensity without considering its direction.
Orthogonal Vectors: Orthogonal vectors are a set of vectors that are perpendicular to each other, meaning they form a right angle (90 degrees) between them. This concept is fundamental in the study of vectors and their applications in various fields, including mathematics, physics, and engineering.
Parallelogram Law: The parallelogram law is a fundamental principle in vector mathematics that describes how two vectors can be combined to form a third vector. It states that the vector sum of two vectors is represented by the diagonal of the parallelogram formed by the two vectors.
Position Vector: A position vector is a mathematical representation of the location of a point in space relative to a reference point or origin. It is a vector quantity that specifies both the direction and the distance from the origin to the point.
Scalar multiplication: Scalar multiplication involves multiplying each entry of a matrix by a constant value, known as the scalar. This operation results in a new matrix where each element is the product of the original element and the scalar.
Scalar Multiplication: Scalar multiplication is the operation of multiplying a vector or matrix by a scalar, which is a single number or quantity. This fundamental operation allows for the scaling or resizing of vectors and matrices, and is a crucial concept in linear algebra and its applications.
Triangle Inequality: The triangle inequality is a fundamental concept in geometry that states the relationship between the lengths of the sides of a triangle. It states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Unit Vector: A unit vector is a vector with a magnitude (length) of 1 that points in a specific direction. It is used to represent the direction of a vector without regard to its magnitude.
Vector: A vector is a mathematical quantity that has both magnitude (size or length) and direction. Vectors are used to represent physical quantities, such as velocity, force, and displacement, which require both a numerical value and a specific direction.
Vector Calculus: Vector calculus, also known as vector analysis, is a branch of mathematics that deals with the differentiation and integration of vector fields. It provides a powerful framework for analyzing and describing the behavior of physical quantities that have both magnitude and direction, such as force, velocity, and electromagnetic fields.
Vector Field: A vector field is a function that assigns a vector to every point in a given space, typically a two-dimensional or three-dimensional space. It is a mathematical concept used to describe the behavior of physical quantities, such as force, velocity, or electric fields, which have both magnitude and direction at each point in a region.
Vector Space: A vector space is a mathematical structure that consists of a collection of objects called vectors, which can be added together and multiplied by scalars, with the operations of vector addition and scalar multiplication satisfying certain axioms. Vector spaces are fundamental in the study of linear algebra and have applications in various fields, including physics, engineering, and computer science.
Velocity Vector: The velocity vector is a mathematical representation of an object's speed and direction of motion. It is a vector quantity, meaning it has both magnitude (speed) and direction, and is used to describe the movement of an object in a specific coordinate system.
Zero Vector: The zero vector is a special vector in vector spaces that has all of its components equal to zero. It represents the origin or starting point of a vector space and has no magnitude or direction.
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