Vectors
804 Basic Vector Operations
Vectors are quantities that have both length (magnitude) and direction.
Examples: electric/magnetic fields, air flow, velocity, acceleration.
13.1 Vectors in the Plane
Vectors in the xy-plane will be discussed and then extended to three dimensions.
Notation for vectors: Vector from P (tail) to Q (head) is denoted as ( \textbf{PQ} ) or similarly using u and v for vectors.
Scalar Multiplication
A scalar ( c ) can be combined with vector ( \mathbf{v} ) using scalar-vector multiplication:
Resulting vector: ( c\mathbf{v} ) (length is ( |c| \cdot |\mathbf{v}| )); same direction if ( c > 0 ), opposite if ( c < 0 ).
Example: ( 3 extbf{v} ) is three times as long, opposite direction for ( -2 extbf{v} ).
Properties of Vectors
Equal vectors must have the same magnitude and direction.
Scalars have magnitude but no direction (e.g., mass, temperature).
Zero vector (\mathbf{0}): has length 0 and no direction.
Vector Addition and Subtraction
For example, consider a plane flying in a crosswind:
Velocity vectors are combined using:
Triangle Rule: Place the tail of one vector at the head of the other.
Parallelogram Rule: Vectors form adjacent sides of a parallelogram; the resultant is the diagonal.
For vector difference: ( \mathbf{u} - \mathbf{v} = \mathbf{u} + (-\mathbf{v}) ).
13.2 Vectors in Three Dimensions
Introduction of the xyz-coordinate system:
Consists of three axes: x, y, z; six types of planes (xy, xz, yz).
Formulation and properties of vectors extend to three dimensions.
Components and Magnitude of Vectors
Vectors can be represented using components in the Cartesian plane:
For vector ( \mathbf{v} ) with tail at the origin: ( \mathbf{v} = (v_1, v_2) ) in 2D or ( \mathbf{v} = (v_1, v_2, v_3) ) in 3D.
Magnitude of a vector is calculated:
Magnitude formula: ( |\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2} ).
Scalar Multiplication and Unit Vectors
A unit vector has a length of 1. Common unit vectors:
( \mathbf{i} = (1, 0, 0) ), ( \mathbf{j} = (0, 1, 0) ), ( \mathbf{k} = (0, 0, 1) ).
Writing any vector as a combination of unit vectors allows us to express direction clearly.
Properties of Vector Operations
Commutative property: ( \mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u} )
Associative property: ( (\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w}) )
Additive identity: ( \mathbf{u} + \mathbf{0} = \mathbf{u} )
Additive inverse: ( \mathbf{u} + (-\mathbf{u}) = \mathbf{0} )
Distributive properties hold for both addition and scalar multiplication.
Velocity Vectors in Crosswinds
The resulting velocity of a boat crossing a river can be calculated using vector addition considering wind velocity.
Frame of reference and direction are crucial to determine resultant motion.
Cross Product
The cross product of vectors results in a new vector that is orthogonal to both specified vectors.
Important properties of the cross product include:
Anticommutative: ( \mathbf{u} \times \mathbf{v} = -\mathbf{v} \times \mathbf{u} )
magnitude: ( |\mathbf{u} \times \mathbf{v}| = |\mathbf{u}| |\mathbf{v}| \sin(\theta) )
A right-hand rule determines the direction of the resultant vector.
Application Example - Torque
Torque from force applied to an object can be found using the cross product, where torque points in the direction perpendicular to both force and radius vector, essential in mechanical applications.