Glossary

1.1 Vector operations and properties

2 min read•august 6, 2024

Vectors are the building blocks of multidimensional math. They help us describe motion, forces, and spatial relationships. In this section, we'll learn how to add, multiply, and manipulate vectors, which is crucial for understanding more complex concepts.

We'll dive into vector operations like addition and multiplication, as well as special products like dot and cross products. We'll also explore vector properties, including unit vectors and . These tools will be essential for solving real-world problems in physics and engineering.

Vector Operations

Adding and Multiplying Vectors

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  • combines two or more vectors to create a new vector ()
    • Graphically, vector addition follows the or
    • Analytically, add the corresponding components of the vectors (v1+v2=<v1x+v2x,v1y+v2y,v1z+v2z>v_1 + v_2 = <v_{1x} + v_{2x}, v_{1y} + v_{2y}, v_{1z} + v_{2z}>)
  • modifies the magnitude of a vector without changing its direction
    • Multiply each component of the vector by the scalar value (kv=<kvx,kvy,kvz>kv = <kv_x, kv_y, kv_z>)
    • Negative scalar values reverse the direction of the vector

Dot Product and Cross Product

  • () calculates the scalar value resulting from the multiplication of two vectors
    • Formula: v1⋅v2=v1xv2x+v1yv2y+v1zv2zv_1 \cdot v_2 = v_{1x}v_{2x} + v_{1y}v_{2y} + v_{1z}v_{2z}
    • Geometrically, v1⋅v2=∣v1∣∣v2∣cos⁡θv_1 \cdot v_2 = |v_1||v_2|\cos\theta, where θ\theta is the angle between the vectors
    • Dot product is commutative: v1⋅v2=v2⋅v1v_1 \cdot v_2 = v_2 \cdot v_1
  • () computes a new vector perpendicular to the plane containing the two input vectors
    • Formula: v1×v2=<v1yv2z−v1zv2y,v1zv2x−v1xv2z,v1xv2y−v1yv2x>v_1 \times v_2 = <v_{1y}v_{2z} - v_{1z}v_{2y}, v_{1z}v_{2x} - v_{1x}v_{2z}, v_{1x}v_{2y} - v_{1y}v_{2x}>
    • Magnitude of the cross product: ∣v1×v2∣=∣v1∣∣v2∣sin⁡θ|v_1 \times v_2| = |v_1||v_2|\sin\theta
    • Direction determined by the right-hand rule
    • Cross product is not commutative: v1×v2=−v2×v1v_1 \times v_2 = -v_2 \times v_1

Vector Properties

Unit Vectors and Magnitude

  • represents the direction of a vector with a magnitude of 1
    • Denoted by a hat (v^\hat{v}) and calculated by dividing a vector by its magnitude: v^=v∣v∣\hat{v} = \frac{v}{|v|}
    • Standard unit vectors: i^=<1,0,0>\hat{i} = <1, 0, 0>, j^=<0,1,0>\hat{j} = <0, 1, 0>, k^=<0,0,1>\hat{k} = <0, 0, 1>
  • (length) is the distance from the initial point to the terminal point of the vector
    • Calculated using the : ∣v∣=vx2+vy2+vz2|v| = \sqrt{v_x^2 + v_y^2 + v_z^2}

Vector Relationships and Decomposition

  • calculates the component of one vector in the direction of another
    • Formula: projv2v1=v1⋅v2∣v2∣2v2proj_{v_2}v_1 = \frac{v_1 \cdot v_2}{|v_2|^2}v_2
    • : compv2v1=v1⋅v2∣v2∣comp_{v_2}v_1 = \frac{v_1 \cdot v_2}{|v_2|}
  • are perpendicular to each other, with a dot product equal to zero (v1⋅v2=0v_1 \cdot v_2 = 0)
  • breaks a vector into its components along specified directions
    • : v=vxi^+vyj^+vzk^v = v_x\hat{i} + v_y\hat{j} + v_z\hat{k}
    • Decomposition along non-standard directions using vector projection

Key Terms to Review (20)

Commutative Property: The commutative property states that the order in which two elements are combined does not affect the result of the operation. This property is fundamental in various mathematical contexts, including vector operations, where addition and multiplication of vectors can be performed in any order without changing the outcome.
Component Form: Component form is a way of representing a vector using its individual components along the coordinate axes. This format clearly indicates the vector's direction and magnitude by expressing it as an ordered pair or triplet, such as (x, y) in two dimensions or (x, y, z) in three dimensions. Understanding component form is crucial for performing vector operations, such as addition, subtraction, and scalar multiplication, as it allows for easy manipulation and calculation of vectors in various applications.
Cross Product: The cross product is a mathematical operation that takes two vectors in three-dimensional space and produces a third vector that is orthogonal (perpendicular) to both of the original vectors. This operation is crucial for determining areas, angles, and orientations in geometry, and it plays a significant role in analyzing the properties of vector fields, normal vectors to surfaces, and surface integrals.
Dot product: The dot product is a mathematical operation that takes two vectors and returns a single scalar value, representing the magnitude of one vector in the direction of another. It provides insight into the relationship between two vectors, including their angle and alignment. The dot product can be calculated using the formula: $$ ext{A} \cdot \text{B} = |\text{A}| |\text{B}| \cos(\theta)$$, where $$\theta$$ is the angle between the two vectors, or by multiplying corresponding components of the vectors and summing them up.
Euclidean norm: The Euclidean norm, often referred to as the length or magnitude of a vector, is a measure of a vector's distance from the origin in Euclidean space. It is calculated using the square root of the sum of the squares of its components, allowing for the geometric interpretation of vectors in multi-dimensional spaces. This concept is crucial for understanding vector operations and properties, as well as analyzing vector fields.
Magnitude: Magnitude refers to the length or size of a vector, representing its overall strength or amount without regard to its direction. In both mathematical and physical contexts, magnitude is crucial because it provides a quantitative measure of a vector’s effect, influencing calculations in operations and properties of vectors as well as visual interpretations in vector fields. Understanding magnitude helps in comprehending how vectors interact and affect systems in two and three dimensions.
Orthogonal Vectors: Orthogonal vectors are vectors that are perpendicular to each other, meaning they meet at a right angle (90 degrees). In the context of vector operations and properties, orthogonality is significant because it implies that the dot product of two orthogonal vectors is zero, which indicates that there is no projection of one vector onto the other. This property is essential in various applications, including geometry, physics, and engineering, where understanding the relationship between different directions is crucial.
Parallelogram Law: The parallelogram law is a geometric principle that relates to vector addition, stating that if two vectors are represented as two adjacent sides of a parallelogram, the resultant vector can be found by drawing a diagonal of the parallelogram. This law helps in understanding how vectors can combine both in terms of magnitude and direction, emphasizing the properties of vector operations and their resultant outcomes.
Rectangular Decomposition: Rectangular decomposition is the process of expressing a vector in terms of its components along the coordinate axes, typically represented as the sum of its horizontal and vertical (or x and y) components. This method allows for easier calculations and analysis of vectors by breaking them down into simpler parts, which can be individually managed and manipulated. Understanding this concept is essential for performing vector operations like addition, subtraction, and determining magnitudes and directions.
Resultant Vector: A resultant vector is the single vector that represents the combined effect of two or more vectors acting simultaneously. It can be found by vector addition, which takes into account both the magnitude and direction of each contributing vector. This concept is fundamental when analyzing forces, velocities, or any quantities that have both size and direction, as it simplifies complex interactions into a single representation.
Scalar Multiplication: Scalar multiplication is the process of multiplying a vector by a scalar (a single number), resulting in a new vector whose direction remains the same if the scalar is positive, but whose magnitude is scaled by the absolute value of the scalar. This operation plays a crucial role in manipulating vectors and can affect their length and orientation in space, making it essential for various applications in vector operations and properties.
Scalar Product: The scalar product, also known as the dot product, is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This operation measures the cosine of the angle between two vectors and reflects how much one vector extends in the direction of another. It's a key concept in vector operations as it links geometric interpretations with algebraic calculations, making it essential for understanding projections and the angle relationships between vectors.
Scalar Projection: Scalar projection refers to the length of the shadow or projection of one vector onto another vector, essentially quantifying how much of one vector lies in the direction of another. This concept is crucial for understanding how vectors relate to each other, as it helps in breaking down vector components along specified directions, which is essential in various vector operations and properties.
Triangle Method: The triangle method is a graphical technique used to visualize vector addition by forming a triangle with two vectors. This method allows for an intuitive understanding of how two vectors combine, leading to the resultant vector that represents their sum. It highlights the concept of vector direction and magnitude while providing a straightforward way to perform operations involving vectors.
Unit Vector: A unit vector is a vector that has a magnitude of one and indicates direction without affecting the scale of the quantity. These vectors are crucial in various calculations as they simplify the representation of directions in vector operations and properties, making them fundamental in mathematics and physics.
Vector Addition: Vector addition is the mathematical operation that combines two or more vectors to produce a resultant vector. This process involves adding the corresponding components of the vectors, and it adheres to specific properties such as commutativity and associativity, making it a fundamental operation in vector mathematics.
Vector Decomposition: Vector decomposition is the process of breaking a vector into its constituent components, typically along specified axes, to simplify calculations and analyses. This method allows for understanding how a vector behaves in different dimensions by expressing it as a sum of its projections on those axes, making it easier to apply various vector operations and properties.
Vector Magnitude: Vector magnitude is a measure of the length or size of a vector, representing how far the vector stretches in space regardless of its direction. It is calculated using the square root of the sum of the squares of its components, often expressed as $$|| extbf{v}|| = ext{sqrt}(v_1^2 + v_2^2 + v_3^2)$$ for a vector \textbf{v} with components \(v_1, v_2, v_3\). Understanding vector magnitude is crucial for performing operations like vector addition, scalar multiplication, and determining physical quantities such as displacement and force.
Vector Product: The vector product, also known as the cross product, is an operation that takes two vectors in three-dimensional space and produces a third vector that is perpendicular to both of the original vectors. This new vector's direction is determined by the right-hand rule, and its magnitude is given by the product of the magnitudes of the two vectors and the sine of the angle between them. The vector product is useful for calculating torque, angular momentum, and determining the area of parallelograms formed by vectors.
Vector Projection: Vector projection is the process of projecting one vector onto another, resulting in a vector that represents the component of the first vector in the direction of the second vector. This operation is crucial for understanding how vectors interact, as it allows us to break down complex vector relationships into simpler components, highlighting their direction and magnitude along specified axes.
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