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Vectors

Irony is wasted on the stupid, Oscar Wilde.

I have yet to see any problem, however complicated, which, when looked at in the right way did not become still more complicated, Paul Anderson.

Definition and Explanation of Vectors

Definition. A vector $\vec{AB}$ is a geometric object or quantity that has both magnitude (or length) and direction. In mathematics, physics, and engineering, vectors are used to describe various physical quantities, such as force, velocity, acceleration, fluid flow, momentum, electric and magnetic fields, etc.

Components and Representation

Vectors in an n-dimensional Euclidean space can be represented as coordinates vectors in a Cartesian coordinate system. For example, in a 3-dimensional space, a vector can be represented using the standard basis vectors $\vec{i},\vec{j},\vec{k}$, which correspond to the x, y, and z axes, respectively. A vector $\vec{A} = a_1\vec{i}+a_2\vec{j}+a_3\vec{k}$ or $\vec{A} = ⟨a_1, a_2, a_3⟩$ represents a vector with components a1, a2, and a3 along the x, y, and z axes respectively.

The point A from where the vector starts is called its initial point, and the point B where it ends is called its terminal point.

The distance between the initial and terminal point of a vector is called its magnitude, denoted as $|\vec{AB}|$ or $||\vec{AB}||$. The magnitude of a vector represents its size or length. The arrow from A to B indicates the direction of the vector, $\vec{AB}$ represents a vector that starts from point A and ends at point B, e.g., $\vec{OA}=⟨3, 2, 1⟩$ or simply $\vec{A}$ starts from the origin of coordinates (0, 0, 0) and points towards the coordinates (3, 2, 1). It has three components, namely 3 along the x-axis, 2 along the y-axis, and 1 along the z-axis.

Image   Definition. The magnitude of the vector $\vec{A} = ⟨a_1, a_2, a_3⟩$, also known as length or norm, is given by the square root of the sum of its components squared, $|\vec{A}|~ or~ ||\vec{A}|| = \sqrt{a_1^2+a_2^2+a_3^2}$, e.g., $||< 3, 2, 1 >|| = \sqrt{3^2+2^2+1^2}=\sqrt{14}$, $||< 3, -4, 5 >|| = \sqrt{3^2+(-4)^2+5^2} = \sqrt{9+ 16 + 25} = \sqrt{50}=5\sqrt{2}$, or $||< 1, 0, 0 >|| = \sqrt{1^2+0^2+0^2}=\sqrt{1}=1$.

Definition. Any vector with a magnitude of 1, i.e., $|\vec{A}|~ or~ ||\vec{A}|| = \sqrt{a_1^2+a_2^2+a_3^2} = 1$, is called a unit vector. It is used to indicate direction without specifying magnitude. The standard basic vectors $\vec{i} = ⟨1, 0, 0⟩,~\vec{j} = ⟨0, 1, 0⟩$, and $\vec{k} = ⟨0, 0, 1⟩$ are all unit vectors because their magnitudes are 1.

Vector Arithmetic

It may be represented graphically by placing the tail of the second vector $\vec{B}$ at the head of the first vector $\vec{A}$, and then drawing an arrow from the tail of the first vector to the head of the second. The new arrow drawn represents the vector $\vec{A+B}$ (Figure ii) and this is called the parallelogram rule because $\vec{A}$ and $\vec{B}$ form the sides of a parallelogram and $\vec{A+B}$ is one of the diagonals.

To subtract $\vec{B}$ from $\vec{A}$, place the tails of both vectors $\vec{A}$ and $\vec{B}$ at the same point, and then draw an arrow from the head of $\vec{B}$ to the head of $\vec{A}$. This new arrow represents the vector $\vec{A}-\vec{B}$(Figure iii).

Intuitively, multiplying a vector by a positive scalar r stretches or compresses this vector by that factor. Geometrically, if $\vec{A}=⟨2, 4, 1⟩$, then $7\vec{A}$ stretches $\vec{A}$ by a factor of 7. If the scalar r is negative, then the vector not only changes its magnitude but also reverses its direction or flips around (by an angle of 180°).

Properties of Vector Addition and Scalar Multiplication

∀$\vec{v},\vec{u},\vec{w}$ vectors, ∀a, b ∈ ℝ:

  1. Commutative property of vector addition states that the order in which you add vectors does not matter, $\vec{v}+\vec{w}=\vec{w}+\vec{v}$.
  2. Associative property of vector addition states that when adding three arbitrary vectors, the result is the same regardless of how the additions are grouped, $\vec{v}+(\vec{u}+\vec{w})=(\vec{v}+\vec{u})+\vec{w}$.
  3. Additive Identity property states that adding the zero vector $\vec{0}$ to any vector $\vec{v}$, leaves the vector unchanged: $\vec{v}+\vec{0}=\vec{v}$.
  4. Multiplicative Identity property states that multiplying a vector $\vec{v}$ by the scalar 1, leaves the vector unchanged, $1\vec{v} = \vec{v}$
  5. Distributive property of scalar multiplication over vector addition states that multiplying a vector sum by a scalar is the same as multiplying each vector by the scalar and then adding the results, $a(\vec{v}+\vec{w})=a\vec{v}+a\vec{w}$.
  6. Distributive property of scalar addition over vector multiplication states that multiplying a vector by the sum of two scalars is the same as multiplying the vector by each scalar separately and then adding the results, $(a+b)\vec{v} = a\vec{v}+b\vec{v}.$

Dot product.

The dot or scalar product is a fundamental operation between two vectors. It produces a scalar quantity that represents the projection of one vector onto another.

Definition. The dot product of two vectors $\vec{A}$ and $\vec{B}$ is the sum of the products of their corresponding components. Mathematically, it is defined as $\vec{A}·\vec{B} = \sum a_ib_i = a_1b_1 + a_2b_2 + a_3b_3.$

Examples: $\vec{A}·\vec{B} = \sum a_ib_i = ⟨2, 2, -1⟩·⟨5, -3, 2⟩ = a_1b_1 + a_2b_2 + a_3b_3 = 2·5+2·(-3)+(-1)·2 = 10-6-2 = 2; \vec{A}·\vec{B} = ⟨3, 5, 2⟩·⟨-1, 3, 0⟩ = 3·(-1)+5·3+2·0 = -3 + 15 + 0 = 12, \vec{A}·\vec{B} = ⟨-3, 2, 2⟩·⟨-4, 2, 1⟩ = (−3)⋅(−4)+ 2⋅2 +2⋅1 = 12 + 4 + 2 = 18.$

Properties

  1. Distributive property of the dot product over vector addition. The dot product of a vector $\vec{u}$ with the sum of two vectors, say $\vec{v}$ and $\vec{w}$, is equal to the sum of the dot products of $\vec{u}$ with $\vec{v}$ and $\vec{w}$ individually: $\vec{u}·(\vec{v}+\vec{w})=\vec{u}·\vec{v}+\vec{u}·\vec{w}$.

    Let $\vec{u}$ = ⟨1, 2, 3⟩, $\vec{v}$ = ⟨4, 5, 6⟩, and $\vec{w}$ = ⟨7, 8, 9⟩, $\vec{u}·(\vec{v}+\vec{w})$ = ⟨1, 2, 3⟩(⟨4, 5, 6⟩+⟨7, 8, 9⟩) = ⟨1, 2, 3⟩⋅⟨11, 13, 15⟩= (1⋅11) + (2⋅13) + (3⋅15) = 11 + 26 + 45 = 82. $\vec{u}·\vec{v}+\vec{u}·\vec{w}$ = ⟨1, 2, 3⟩⋅⟨4, 5, 6⟩ + ⟨1, 2, 3⟩⋅⟨7, 8, 9⟩ = ((1⋅4) + (2⋅5) + (3⋅6)) + ((1⋅7) + (2⋅8) + (3⋅9)) = (4 + 10 + 18) + (7+ 16 + 27) = 32 + 50 = 82

  2. Scalar multiplication property. The dot product of a scalar multiple of a vector $\vec{v}$ with another vector $\vec{w}$ is equal to the scalar multiple of the dot product of $\vec{v}$ and $\vec{w}, (c\vec{v})·\vec{w} = \vec{v}·(c\vec{w})=c·(\vec{v}·\vec{w})$

  3. Commutative property of the dot product, the order of the vectors does not affect the result, $\vec{v}·\vec{w}=\vec{w}·\vec{v}$.

  4. Dot product of any vector with the zero vector is always zero, $\vec{u}·\vec{0} = 0$.

  5. The dot product of a vector with itself is equal to the square of its magnitude, $\vec{u}·\vec{u}=|\vec{u}|^2.$ Let $\vec{u} = ⟨3, 4, 5⟩, \vec{u}·\vec{u}= ⟨3, 4, 5⟩⋅⟨3, 4, 5⟩=(3⋅3) + (4⋅4) + (5⋅5) = 9 + 16 + 25 = 50. |\vec{u}|=\sqrt{3^2+4^2+5^2}=\sqrt{50}, |\vec{u}|^2=50.$

  6. If the dot product of a vector with itself is zero, then the vector must be the zero vector. $\text{If}~ \vec{u}·\vec{u} = 0⇒ \vec{u} = \vec{0}$.

These properties are essential for various applications in physics, engineering, and computer science, providing a foundation for understanding vector projections, angles between vectors, and more.

Solved exercises

Dot Product

$3\vec{a}-4\vec{b} = 3⟨3,2,−1⟩-4⟨0,6,7⟩ = ⟨3⋅3,3⋅2,3⋅−1⟩-⟨4⋅0,4⋅6,4⋅7⟩ = 9,6,−3⟩−⟨0,24,28⟩=⟨9−0,6−24,−3−28⟩=⟨9,−18,-31⟩$

$|\vec{w}| = \sqrt{5^2 + 4^2 + 1^2}=\sqrt{42}$

$\hat{\mathbf{w}}=\frac{1}{|\vec{w}|}|\vec{w} = \sqrt{42}·⟨5, 4, 1⟩= ⟨\frac{5}{\sqrt{42}},\frac{4}{\sqrt{42}},\frac{1}{\sqrt{42}}⟩$

Bibliography

This content is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License and is based on MIT OpenCourseWare [18.01 Single Variable Calculus, Fall 2007].
  1. NPTEL-NOC IITM, Introduction to Galois Theory.
  2. Algebra, Second Edition, by Michael Artin.
  3. LibreTexts, Calculus and Calculus 3e (Apex). Abstract and Geometric Algebra, Abstract Algebra: Theory and Applications (Judson).
  4. Field and Galois Theory, by Patrick Morandi. Springer.
  5. Michael Penn, and MathMajor.
  6. Contemporary Abstract Algebra, Joseph, A. Gallian.
  7. YouTube’s Andrew Misseldine: Calculus. College Algebra and Abstract Algebra.
  8. MIT OpenCourseWare 18.01 Single Variable Calculus, Fall 2007 and 18.02 Multivariable Calculus, Fall 2007.
  9. Calculus Early Transcendentals: Differential & Multi-Variable Calculus for Social Sciences.
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