Vectors in 3D space

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.

While scalars only have a magnitude (like temperature, mass, or time), vectors describe physical quantities where the direction is essential. In physics and engineering, vectors represent forces, velocity, acceleration, fluid flow, momentum, and magnetic fields.

Components and Representation

Vectors in an $n$-dimensional Euclidean space are usually represented as coordinate vectors in a Cartesian system.

In a 3-dimensional space, a vector is expressed using the standard basis vectors $\vec{i}$, $\vec{j}$, and $\vec{k}$, which correspond to the $x$, $y$, and $z$ axes respectively. A vector can be written as: $\vec{A} = a_1\vec{i} + a_2\vec{j} + a_3\vec{k} \quad \text{or} \quad \vec{A} = \langle a_1, a_2, a_3 \rangle$

This represents a vector with components $a_1$, $a_2$, and $a_3$ stretching along the $x$, $y$, and $z$ axes.

• Initial Point (Tail): The point $A$ where the vector starts.
• Terminal Point (Head): The point $B$ where the vector ends.

If a vector starts at the origin $(0, 0, 0)$ and ends at the coordinates $(3, 2, 1)$, it is denoted as $\vec{A} = \langle 3, 2, 1 \rangle$.

Magnitude and Unit Vectors

The distance between the initial and terminal point of a vector is 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., $\vec{OA}=\vert{}\langle 3, 2, 1\vert{}\rangle$ 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 is found using the 3D distance formula (Pythagorean theorem), that is, the square root of the sum of its components squared, $\vert{}\vec{A}\vert{}~ or~ \vert{}\vert{}\vec{A}\vert{}\vert{} = \sqrt{a_1^2+a_2^2+a_3^2}$, e.g., $\vert{}\vert{}\langle 3, 2, 1 \vert{}\rangle\vert{} = \sqrt{3^2+2^2+1^2}=\sqrt{14}$, $||< 3, -4, 5 >|| = \sqrt{3^2+(-4)^2+5^2} = \sqrt{9+ 16 + 25} = \sqrt{50}=5\sqrt{2}$, 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

• The sum of two vectors is found by adding their corresponding components. This can be represented mathematically as: $\vec{A}+ \vec{B} = (a_1+b_1)\vec{i}+(a_2+b_2)\vec{j}+(a_3+b_3)\vec{k} = \langle a_1+b_1, a_2+b_2, a_3+b_3 \rangle$, e.g., $\vec{A}+\vec{B} = \langle 3, 2 \rangle + \langle 5, 6 \rangle = \langle 8, 8 \rangle$, $\vec{A}+\vec{B} = ⟨2,−1,3⟩ + ⟨−3,4,1⟩ = ⟨−1,3,4⟩, \vec{A}+\vec{B} = <3, 3, 3> + <1, 0, −1> = <4, 3, 2>$.

• 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): $\vec{A} - \vec{B} = (a_1-b_1)\vec{i}+(a_2-b_2)\vec{j}+(a_3-b_3)\vec{k} = \langle a_1-b_1, a_2-b_2, a_3-b_3 \rangle$, e.g., $\vec{A}-\vec{B} = ⟨2,−1,3⟩ - ⟨−3,4,1⟩ = ⟨5, -5, 2⟩, \vec{A}-\vec{B} = \langle 6, -2, -5 \rangle - \langle -4, 7, -8 \rangle = \langle 10, -9, 3 \rangle, \vec{B}-\vec{A} = ⟨-4, 7, -8⟩ - ⟨6, -2, -5⟩ = ⟨-10, 9, -3⟩.$

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 $\vec{B}$ together, 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).

• Scalar multiplication is the multiplication of a vector by a scalar, a real number, changing its magnitude without altering its direction. This operation is performed by multiplying each component of the vector by the scalar value, $c\vec{A} = (ca_1)\vec{i}+(ca_2)\vec{j}+(ca_3)\vec{k} = \langle ca_1, ca_2, ca_3 \rangle$, e.g., $2\vec{A} = 2⟨3, 1, 1⟩ = ⟨6, 2, 2⟩; -3\vec{A} = -3⟨3, 2, -1⟩= (-9)\vec{i}+(-6)\vec{j}+(3)\vec{k} = < -9, -6, 3>$.

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, $\forall r \in \mathbb{R}$:

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.

There are two ways to define the dot product:

Algebraic 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 $\vec{A} \cdot \vec{B} = a_1b_1 + a_2b_2 + a_3b_3.$

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

Geometric Definition: The product of their magnitudes multiplied by the cosine of the angle $\theta$ between them. $\vec{A} \cdot \vec{B} = \vert{}\vec{A}\vert{}\vert{}\vec{B}\vert{}\cos(\theta)$
The geometric definition gives us a simple test for perpendicularity. If two non-zero vectors are perpendicular ($90^\circ$), $\cos(90^\circ) = 0$, meaning their dot product will be exactly $0$.

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 equals its magnitude squared, $\vec{u}·\vec{u}=|\vec{u}|^2.$ Let $\vec{u} = ⟨3, 4, 5⟩, \vec{u}·\vec{u}= ⟨3, 4, 5⟩⋅⟨3, 4, 5⟩=(3⋅3) + (4⋅4) + (5⋅5) = 9 + 16 + 25 = 50.$ Notice that $|\vec{u}|=\sqrt{3^2+4^2+5^2}=\sqrt{50}, |\vec{u}|^2=50.$

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

Plot the points (3, 2, 5) and (4, -2, -1) on the axes below. Figure 1.

To plot points in a 3D Cartesian coordinate system, you move sequentially along the $x$, $y$, and $z$ axes:

• Point (3, 2, 5): Start at the origin (0,0,0). Move 3 units forward along the positive $x$-axis, 2 units right parallel to the positive $y$-axis, and finally 5 units up parallel to the positive $z$-axis.
• Point (4, -2, -1): Start at the origin. Move 4 units forward along the positive $x$-axis, 2 units left parallel to the negative $y$-axis, and 1 unit down parallel to the negative $z$-axis.

Plotting 3D Surfaces

When an equation in 3D space is missing one of the variables, the 2D curve extends infinitely along the axis of the missing variable, creating a “cylinder” (in the broad mathematical sense).

Surface 1: $y = x^2$ (Figures 2). In the 2D $xy$-plane, this is a standard parabola. Because $z$ is not restricted in the equation, this parabola is “extruded” infinitely up and down along the $z$-axis. This forms a parabolic cylinder.

Surface 2: $x^2 + z^2 = 16$ (Figures 3). In the 2D $xz$-plane, this is the equation of a circle centered at the origin with a radius of $r = \sqrt{16} = 4$. Because $y$ is missing, this circle extends infinitely along the $y$-axis. This forms a standard circular cylinder.

Vector Arithmetic

Calculate $3\vec{a}-4\vec{b}$ given $\vec{a} = \langle 3, 2, -1 \rangle$ and $\vec{b} = \langle 0, 6, 7 \rangle$.

Solution: First, distribute the scalars into each vector, then subtract the corresponding components:

$3\vec{a} - 4\vec{b} = 3\langle 3, 2, -1 \rangle - 4\langle 0, 6, 7 \rangle = \langle 3 \cdot 3, 3 \cdot 2, 3 \cdot -1 \rangle - \langle 4 \cdot 0, 4 \cdot 6, 4 \cdot 7 \rangle$

$= \langle 9, 6, -3 \rangle - \langle 0, 24, 28 \rangle = \langle 9 - 0, 6 - 24, -3 - 28 \rangle = \langle 9, -18, -31 \rangle$

Finding a Unit Vector

Find a unit vector $\hat{\mathbf{w}}$ in the direction of $\vec{w} = \langle 5, 4, 1 \rangle$.

Solution:

1. Recall. A unit vector is found by dividing a vector by its own magnitude.
2. Find the magnitude (length) of $\vec{w}$: $\vert{}\vec{w}\vert{} = \sqrt{x^2 + y^2 + z^2} = \sqrt{5^2 + 4^2 + 1^2} = \sqrt{25 + 16 + 1} = \sqrt{42}$.
3. Multiply the vector by the reciprocal of the magnitude: $\hat{\mathbf{w}} = \frac{1}{\vert{}\vec{w}\vert{}}\vec{w} = \frac{1}{\sqrt{42}}\langle 5, 4, 1 \rangle = \left\langle \frac{5}{\sqrt{42}}, \frac{4}{\sqrt{42}}, \frac{1}{\sqrt{42}} \right\rangle$

Bibliography

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).
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5. Michael Penn, and MathMajor.
6. Contemporary Abstract Algebra, Joseph, A. Gallian.
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