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. 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.
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 a_{1}, a_{2}, and a_{3} 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.
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.
The sum of two vectors is found by adding their corresponding components. This can be represented mathematically as: $\vec{A}+ \vec{B} = (a_1+b_1)\vec{i}+(a_2+b_2)\vec{j}+(a_3+b_3)\vec{k}$ = < (a_{1}+b_{1}), (a_{2}+b_{2}), (a_{3}+b_{3}) >, e.g., $\vec{A}+\vec{B} = ⟨3, 2⟩ + ⟨5, 6⟩ = ⟨8, 8⟩, \vec{A}+\vec{B} = ⟨2,−1,3⟩ + ⟨−3,4,1⟩ = ⟨−1,3,4⟩, \vec{A}+\vec{B} = <3, 3, 3> + <1, 0, −1> = <4, 3, 2>$.
The subtraction of two vectors is similar to addition and is also done component-wise, it is given by simply subtracting their corresponding components (x, y, and z in 3D): $\vec{A} - \vec{B} = (a_1-b_1)\vec{i}+(a_2-b_2)\vec{j}+(a_3-b_3)\vec{k}$ = < (a_{1}-b_{1}), (a_{2}-b_{2}), (a_{3}-b_{3}) >, e.g., $\vec{A}-\vec{B} = ⟨2,−1,3⟩ - ⟨−3,4,1⟩ = ⟨5, -5, 2⟩, \vec{A}-\vec{B} = ⟨6, -2, -5⟩ - ⟨-4, 7, -8⟩ = ⟨10, -9, 3⟩, \vec{B}-\vec{A} = ⟨-4, 7, -8⟩ - ⟨6, -2, -5⟩ = ⟨-10, 9, -3⟩.$
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°).
∀$\vec{v},\vec{u},\vec{w}$ vectors, ∀a, b ∈ ℝ:
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: $\vec{A}·\vec{B} = \sum a_ib_i = ⟨2, 2, -1⟩·⟨5, -3, 2⟩ = a_1b_1 + a_2b_2 + a_3b_3 = 2·5+2·(-3)+(-1)·2 = 10-6-2 = 2; \vec{A}·\vec{B} = ⟨3, 5, 2⟩·⟨-1, 3, 0⟩ = 3·(-1)+5·3+2·0 = -3 + 15 + 0 = 12, \vec{A}·\vec{B} = ⟨-3, 2, 2⟩·⟨-4, 2, 1⟩ = (−3)⋅(−4)+ 2⋅2 +2⋅1 = 12 + 4 + 2 = 18.$
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
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 $\vec{w}, (c\vec{v})·\vec{w} = \vec{v}·(c\vec{w})=c·(\vec{v}·\vec{w})$
Commutative property of the dot product, the order of the vectors does not affect the result, $\vec{v}·\vec{w}=\vec{w}·\vec{v}$.
Dot product of any vector with the zero vector is always zero, $\vec{u}·\vec{0} = 0$.
The dot product of a vector with itself is equal to the square of its magnitude, $\vec{u}·\vec{u}=|\vec{u}|^2.$ Let $\vec{u} = ⟨3, 4, 5⟩, \vec{u}·\vec{u}= ⟨3, 4, 5⟩⋅⟨3, 4, 5⟩=(3⋅3) + (4⋅4) + (5⋅5) = 9 + 16 + 25 = 50. |\vec{u}|=\sqrt{3^2+4^2+5^2}=\sqrt{50}, |\vec{u}|^2=50.$
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.
Plot the surfaces y = x^{2} and x^{2} + z^{2} = 16. Sol: Figures 2 and 3 respectively.
Calculate $3\vec{a}-4\vec{b}$ where $\vec{a} = ⟨3, 2, -1⟩, \vec{b} = ⟨0, 6, 7⟩$.
$3\vec{a}-4\vec{b} = 3⟨3,2,−1⟩-4⟨0,6,7⟩ = ⟨3⋅3,3⋅2,3⋅−1⟩-⟨4⋅0,4⋅6,4⋅7⟩ = 9,6,−3⟩−⟨0,24,28⟩=⟨9−0,6−24,−3−28⟩=⟨9,−18,-31⟩$
$|\vec{w}| = \sqrt{5^2 + 4^2 + 1^2}=\sqrt{42}$
$\hat{\mathbf{w}}=\frac{1}{|\vec{w}|}|\vec{w} = \sqrt{42}·⟨5, 4, 1⟩= ⟨\frac{5}{\sqrt{42}},\frac{4}{\sqrt{42}},\frac{1}{\sqrt{42}}⟩$