Vector Space

Definition: Vector Space @vector-space A vector space over a field $F$ is a set $V$ together with two operations: - Vector addition: $V \times V \to V$ - Scalar multiplication: $F \times V \to V$ satisfying the following axioms: $(V, +)$ is an abelian group Scalar multiplication is associative: $a(bv) = (ab)v$ Distributive laws hold Identity: $1v = v$ for all $v \in V$ Referenced by (3 direct, 32 transitive)Direct references:ScalarVectorTooltip Test PageTransitive (depth 1):remark-9remark-16Hessian MatrixincompressibleBound vectorDirectioninvariance-of-curlJacobian MatrixCross ProductUnit Vectornote-5Surface Normal Vectortheorem-19remark-45Free vectorremark-32Vector Multiplication by a Scalarnote-3Transitive (depth 2):Vector Equalityproof-of-theorem-19remark-46Normal DerivativeDirectional DerivativeZero Vectorremark-30directional-derivative-is-inner-product-of-vector-and-gradremark-23Tangent Vectorgradient-as-surface-normal-vectorproof-of-gradient-as-surface-normal-vectorTransitive (depth 3):Stoke's Theoremgrad-div-curl-related

Scalars

Definition: Scalar @scalar A scalar is an element of a field used to define a vector space . Referenced by (4 direct) Direct references: note-3 Vector Multiplication by a Scalar remark-32 Hessian Matrix

Note @note-3 Typically, especially in physics, a scalar is simply a number, especially a real number .

Vectors

Definition: Vector @vector A vector is an element in a vector space . Referenced by (15 direct, 14 transitive) Direct references: note-5 Free vector Bound vector Direction Unit Vector Cross Product remark-9 remark-16 theorem-19 Surface Normal Vector incompressible invariance-of-curl Hessian Matrix Jacobian Matrix remark-45 Transitive (depth 1): Vector Equality proof-of-theorem-19 remark-46 Normal Derivative Directional Derivative Zero Vector remark-30 directional-derivative-is-inner-product-of-vector-and-grad remark-23 Tangent Vector gradient-as-surface-normal-vector proof-of-gradient-as-surface-normal-vector Transitive (depth 2): Stoke's Theorem grad-div-curl-related

Note @note-5 Typically, in physics and many other applications, when we say vector we mean a vector in $\mathbb{R}^n,$ which is an ordered tuple of real numbers, $\vec{x} = (x_1, x_2, \cdots, x_n).$ This is the algebraic representation of vector $\vec{x}$. The geometric representation of vector $\vec{x}$ is, in an $n$ dimensional space, an arrow or directed line segment. When starting from the origin, it would be a line segment from $\vec{0}$ to the point $(x_1, x_2, \cdots, x_n)$. Physically, a vector has a magnitude and a direction.

Definition: Free vector @free-vector A vector whose endpoints are not fixed at particular points is called a free vector.

Definition: Bound vector @bound-vector A vector with a fixed endpoint is called a bound vector.

Properties of Vectors

Definition: Magnitude (also: norm, length) @magnitude Let $\vec{x} = (x_1, x_2, \cdots, x_n) \in \mathbb{R}^n$. The magnitude, or length, $\vec{x}$ is denoted as $|| \vec{x} ||$ and is defined as: $$ ||\vec{x}|| = \sqrt{ {x_1}^2 + {x_2}^2 + \cdots + {x_n}^2 } = \sqrt{\vec{x} \cdot \vec{x}} = \sqrt{\sum_{i=1}^n x_i^2} $$ Referenced by (2 direct)Direct references:Vector Differential Calculusproof-of-theorem-19

Note @note-9 This is essentially the Pythagorean theorem in $n$ dimensions; in $\mathbb{R}^2$ the magnitude of a vector corresponds to the length of the hypotenuse of a right triangle whose other sides are of length $x_1$ and $x_2.$ While we define norm to be equivalent to magnitude here, this is actually a special case of the more general concept of a norm - there are other norms we could define on $\mathbb{R}^n,$ but I don't have need to explore that yet.

Definition: Direction @direction The direction of a vector can be specified by the angle between it and some fixed reference, such as the $x$-axis. Referenced by (14 direct, 4 transitive) Direct references: Zero Vector Vector Equality Cross Product Vector Differential Calculus Directional Derivative directional-derivative-is-inner-product-of-vector-and-grad proof-of-theorem-19 theorem-19 Surface Normal Vector Normal Derivative remark-30 invariance-of-curl remark-45 remark-46 Transitive (depth 1): remark-23 gradient-as-surface-normal-vector proof-of-gradient-as-surface-normal-vector grad-div-curl-related

Definition: Zero Vector @zero-vector The zero vector $(0, 0, \cdots, 0)$ is denoted as $\vec{0}$ and has no direction. Referenced by (2 direct)Direct references:gradient-as-surface-normal-vectorgrad-div-curl-related

Definition: Vector Equality @vector-equality Two vectors are equal if they have the same coordinates (or equivalently, the same length and direction).

Definition: Unit Vector @unit-vector If a vector has a length of 1, i.e. if $||x|| = 1$, we say the vector is a unit vector . Referenced by (4 direct, 4 transitive) Direct references: Tangent Vector Directional Derivative directional-derivative-is-inner-product-of-vector-and-grad proof-of-theorem-19 Transitive (depth 1): Stoke's Theorem remark-45 Normal Derivative Transitive (depth 2): remark-30

Definition: Vector Addition @vector-addition We perform vector addition by adding two vectors $\vec{x} = (x_1, x_2, \cdots, x_n)$ and $\vec{y} = (y_1, y_2, \cdots, y_n)$ according to the following rule: $$ \vec{x} + \vec{y} = (x_1 + y_1, x_2 + y_2, \cdots, x_n + y_n) $$ that is, by making a new vector where the coordinates are the sums of the respective coordinates in the vectors being summed. Referenced by (1 direct)Direct references:note-15

Note @note-15 Geometrically, vector addition connects vectors head to tail.

Definition: Vector Subtraction @vector-subtraction Similarly, vector subtraction can be performed as: $$ \vec{x} - \vec{y} = (x_1 - y_1, x_2 - y_2, \cdots, x_n - y_n) $$

Multiplication

Definition: Vector Multiplication by a Scalar @vector-multiplication-by-a-scalar We can multiply vectors by a scalar. Given the scalar $c$: $$ c \vec{x} = (c x_1, c x_2, \cdots, c x_n) $$

Definition: Inner product (also: Dot Product, Scalar product of two vectors) @inner-product If $\vec{x}$ and $\vec{y}$ are vectors in $\mathbb{R}^n,$ then their inner product is defined as $$ \vec{X} \cdot \vec{y} = \sum_{i = 1}^n x_i y_i. $$ Referenced by (2 direct, 1 transitive)Direct references:theorem-7-remarkdirectional-derivative-is-inner-product-of-vector-and-gradTransitive (depth 1):proof-of-theorem-19

Theorem @cos-characterization-of-dot-product The dot product of $\vec{u}$ and $\vec{v}$ is $$ \vec{u} \cdot \vec{v} = |\vec{u}||\vec{v}|\cos{\theta} $$ where $\theta$ is the angle between $\vec{u}$ and $\vec{v}.$ Referenced by (1 direct)Direct references:proof-of-theorem-19

Definition: Perpendicular (also: orthogonal) @perpendicular Two vectors $\vec{u}$ and $\vec{v}$ are said to be perpendicular or orthogonal if the angle between them is $\pi/2$ radians, or, equivalently, if the inner product $\vec{u} \cdot \vec{v} = 0.$ Referenced by (2 direct, 4 transitive)Direct references:Cross ProductSurface NormalTransitive (depth 1):Surface Normal Vectorremark-23Transitive (depth 2):gradient-as-surface-normal-vectorproof-of-gradient-as-surface-normal-vector

Definition: Parallel @parallel Two vectors $\vec{x}$ and $\vec{y}$ are said to be parallel if $\vec{x}$ is a scalar multiple of $\vec{y}$, i.e., if there exists some scalar $c$ where $\vec{x} = c \vec{y}.$ Referenced by (1 direct)Direct references:proof-of-theorem-19

Definition: Cross Product (also: Vector Product, Outer Product) @cross-product The cross product $\vec{a} \times \vec{b}$ (read "a cross b") of two vectors $\vec{a}$ and $\vec{b}$ is the vector $\vec{v}$ denoted by $$ \vec{v} = \vec{a} \times \vec{b}. $$ I. If $\vec{a} = \vec{0}$ or $\vec{b} = \vec{0},$ then we define $\vec{v} = \vec{a} \times \vec{b} = \vec{0}.$ II. If both vectors are nonzero vectors, then vector $\vec{v}$ has the length $$ \tag{1} |\vec{v}| = |\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin{\theta}, $$ where $\theta$ is the angle between $\vec{a}$ and $\vec{b}.$ Furthermore, $\vec{a}$ and $\vec{b}$ form the sides of a parallelogram on a plane in space. The area of this parallelogram is precisely given by (1), such that the length $|\vec{v}|$ of the vector $\vec{v}$ is equal to the area of the parallelogram. III. If $\vec{a}$ and $\vec{b}$ lie in the same straight line, i.e. $\vec{a}$ and $\vec{b}$ have the same or opposite directions, then $\theta$ is $0 \degree$ or $180 \degree$ so that $\sin \theta = 0.$ In that case $|\vec{v}| = 0,$ so that $\vec{v} = \vec{a} \times \vec{b} = \vec{0}. $$ IV. If cases I and III do not occur, then $\vec{v}$ is a nonzero vector. The direction of $\vec{v} = \vec{a} \times \vec{b}$ is perpendicular to both $\vec{a}$ and $\vec{b}$ such that $\vec{a}, \vec{b}, \vec{v},$ precisely in this order, form a right-handed triple Referenced by (1 direct)Direct references:remark-23

Remark @remark-23 If $\vec{a} = (a_1, a_2, a_3)$ and $\vec{b} = (b_1, b_2, b_3),$ then we use the symbolic determinant to find the cross product, in this fashion: $$ \vec{v} = \vec{a} \times \vec{b} = \begin{vmatrix}\vec{i} & \vec{j} & \vec{k}\\a_1 & a_2 & a_3\\b_1 & b_2 & b_3\end{vmatrix} = \begin{vmatrix} a_2 & a_3 \\ b_2 & b_3 \end{vmatrix} \vec{i} - \begin{vmatrix} a_1 & a_3 \\ b_1 & b_3 \end{vmatrix} \vec{j} + \begin{vmatrix} a_1 & a_2 \\ b_1 & b_2 \end{vmatrix} \vec{k}. $$

Definition: Linear Combination @linear-combination Let $\vec{v_1}, \vec{v_2}, \cdots, \vec{v_n} \in \mathbb{R}^n$ and $c_1, c_2, \cdots, c_3 \in \mathbb{R}.$ Then, the vector $$ \vec{v} = c_1 \vec{v_1} + c_2 \vec{v_2} + \cdots + c_n \vec{v_n} $$ is called a linear combination of $\vec{v_1}, \vec{v_2}, \cdots, \vec{v_n}.$ Let $\vec{v_1}, \vec{v_2}, \cdots, \vec{v_n} \in \mathbb{R}^n.$ The set of all linear combinations of $\vec{v_1}, \vec{v_2}, \cdots, \vec{v_n}$ is called their span, denoted $\Span{\left(\vec{v_1}, \vec{v_2}, \cdots, \vec{v_n}\right)}.$ That is: $$ \Span{\left(\vec{v_1}, \vec{v_2}, \cdots, \vec{v_n}\right)} = \left\{ \vec{v} \in \mathbb{R}^n : \vec{v} = c_1 \vec{v_1} + c_2 \vec{v_2} + \cdots + c_n \vec{v_n} ~ \text{for some scalars} ~ c_1, c_2, \cdots, c_n \right\} $$