Comprehensive Engineering Study Notes on Vectors

Introduction to Physical Quantities: Scalars and Vectors

• Definition of Scalars: A scalar is a physical quantity that can be described entirely by a single number representing its magnitude. Scalars can be positive, negative, or zero.

  • Examples of Scalars: Distance ($m$), mass ($kg$), temperature ($K$), pressure ($Pa$), work ($J$), energy ($J$), speed ($m\,s^{-1}$), and voltage ($V$).

• Definition of Vectors: A vector is a quantity that requires both a magnitude (size or strength) and a direction for a complete description. Vectors must be manipulated using specific rules of vector algebra.

  • Examples of Vectors: Displacement ($m$), force ($N$), velocity ($m\,s^{-1}$), acceleration ($m\,s^{-2}$), momentum, and electric or magnetic fields.

• Technical Distinctions in Engineering:

  • Speed vs. Velocity: Speed is a scalar (e.g., $40\,km/h$). Velocity is a vector (e.g., $20\,m/s$ due north).

  • Mass vs. Weight: Mass is a scalar (amount of substance, measured in $kg$). Weight is a vector force resulting from gravity, directed vertically downwards (measured in $N$).

  • Distance vs. Displacement: Distance is a scalar representing the total path length. Displacement is "directed distance," representing the change in position from a starting point $A$ to an end point $B$ in a specific direction ($\vec{AB}$).

Mathematical Description and Representation of Vectors

• Directed Line Segments: A vector is represented graphically by a line. Its length corresponds to the magnitude (given a specific scale), and its orientation and arrow indicate the direction.

• Notation Systems:

  • Vector from point $A$ to point $B$: $\vec{AB}$.

  • Standard notation: Boldface (e.g., $\mathbf{a}$) or an underlined letter (e.g., $\underline{a}$).

  • Magnitude notation: Using modulus signs $|\vec{AB}|$ or $|\mathbf{a}|$, or simply the letter without formatting ($a$).

• Properties of Vectors:

  • Equality: Two vectors are equal if they have the identical magnitude and direction, regardless of their position in space. These are often called Free Vectors.

  • Negative Vectors: The vector $-\mathbf{a}$ has the same magnitude as $\mathbf{a}$ but acts in the exact opposite direction. Geometrically, if $\mathbf{a} = \vec{AB}$, then $-\mathbf{a} = \vec{BA}$.

Vector Addition and Subtraction

• The Triangle Law of Addition: To add vector $\mathbf{b}$ to vector $\mathbf{a}$, translate $\mathbf{b}$ without changing its direction or length until its tail coincides with the head of $\mathbf{a}$. The sum (resultant) $\mathbf{c} = \mathbf{a} + \mathbf{b}$ is the vector forming the third side of the triangle from the tail of $\mathbf{a}$ to the head of $\mathbf{b}$.

• Algebraic Rules of Addition:

  • Commutative Law: $\mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a}$.

  • Associative Law: $\mathbf{a} + (\mathbf{b} + \mathbf{c}) = (\mathbf{a} + \mathbf{b}) + \mathbf{c}$.

• Resultant of Concurrent Forces: Forces acting through the same point can be combined into a single resultant force $\mathbf{R}$ using the triangle law.

• Vector Subtraction: Performed by adding the corresponding negative vector. To find $\mathbf{a} - \mathbf{b}$, calculate $\mathbf{a} + (-\mathbf{b})$.

• Multiplying a Vector by a Scalar: If $k$ is a positive scalar, $k\mathbf{a}$ is a vector in the same direction, $k$ times as long. If $k$ is negative, the direction is reversed.

  • Distributive Laws: $k(\mathbf{a} + \mathbf{b}) = k\mathbf{a} + k\mathbf{b}$ and $(k + l)\mathbf{a} = k\mathbf{a} + l\mathbf{a}$.

  • Scaling: $k(l)\mathbf{a} = (kl)\mathbf{a}$.

Resolution and Unit Vectors

• Resolution of Vectors: A single vector can be replaced by two (or more) perpendicular components. For a force $\mathbf{F}$ at angle $\theta$ to the horizontal:

  • Horizontal component: $F \cos(\theta)$.

  • Vertical component: $F \sin(\theta)$.

• Unit Vectors: A vector with a magnitude of $1$. A unit vector in the direction of $\mathbf{a}$ (denoted $\hat{\mathbf{a}}$) is found by dividing the vector by its magnitude: $\hat{\mathbf{a}} = \frac{\mathbf{a}}{|\mathbf{a}|}$.

Cartesian Components of Vectors

• Standard Basis Vectors:

  • $\mathbf{i}$: Unit vector in the direction of the positive x-axis.

  • $\mathbf{j}$: Unit vector in the direction of the positive y-axis.

  • $\mathbf{k}$: Unit vector in the direction of the positive z-axis.

• Two-Dimensional Representation: Any vector $\mathbf{r}$ in the xy-plane is expressed as $\mathbf{r} = a\mathbf{i} + b\mathbf{j}$. The values $a$ and $b$ are the components.

• Column Vector Notation: $\mathbf{r} = \begin{pmatrix} a \\ b \end{pmatrix}$.

• Three-Dimensional Representation: $\mathbf{r} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k}$, corresponding to coordinates $(a, b, c)$.

• Position Vectors: Fixed vectors denoting the position of a point relative to the origin $O$. The position vector of point $P(a, b, c)$ is $\vec{OP} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k}$. Unlike free vectors, position vectors cannot be translated.

• Calculations with Cartesian Forms:

  • Addition: $\mathbf{a} + \mathbf{b} = (a_x + b_x)\mathbf{i} + (a_y + b_y)\mathbf{j} + (a_z + b_z)\mathbf{k}$.

  • Modulus (Length): In 2D, $|\mathbf{r}| = \sqrt{a^2 + b^2}$. In 3D, $|\mathbf{r}| = \sqrt{a^2 + b^2 + c^2}$.

• Vector between two points: If points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$, the vector $\vec{AB} = \mathbf{b} - \mathbf{a}$.

Engineering Applications

• Doppler Effect and Radar Guns: Radar guns measure the velocity component of a vehicle directly toward the gun. If the gun is used at an angle $\theta$ to the traffic line, it measures $v \cos(\theta)$. To maintain an error within $5\%$, reading at a specific distance $d$ is required. For a gun $10\,m$ to the side, taking $|v| \cos(\theta) = 0.95|v|$ yields $\cos(\theta) = 0.95$ and $d = \frac{10}{\sqrt{1 - (0.95)^2}} \approx 32.03\,m$.

• Aeroplane Steady Flight: The forces involved are Thrust ($T$), Drag ($D$), Weight ($W$), and Lift ($L$).

  • Equilibrium along path: $T \cos(\alpha) - D - W \sin(\beta) = 0$.

  • Equilibrium perpendicular to path: $T \sin(\alpha) + L - W \cos(\beta) = 0$.

  • Example: For mass $72000$ tonnes, $D = 130\,kN$, $L = 690\,kN$, and $\beta = 6^{\circ}$, calculating steady flight requires $T = 204210\,N$ and $\alpha = 3.500^{\circ}$.

The Scalar Product (Dot Product)

• Definition: The scalar product of $\mathbf{a}$ and $\mathbf{b}$ is defined as: $\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos(\theta)$, where $\theta$ is the angle ($0 \le \theta \le \pi$) between the vectors when their tails coincide.

• Key Rules:

  • Commutative: $\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}$.

  • Basis vectors: $\mathbf{i} \cdot \mathbf{i} = \mathbf{j} \cdot \mathbf{j} = \mathbf{k} \cdot \mathbf{k} = 1$ and $\mathbf{i} \cdot \mathbf{j} = \mathbf{j} \cdot \mathbf{k} = \mathbf{k} \cdot \mathbf{i} = 0$.

  • Perodicity: The scalar product of perpendicular vectors is zero.

  • Magnitude: $|\mathbf{r}|^2 = \mathbf{r} \cdot \mathbf{r}$.

• Cartesian Formula: If $\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k}$ and $\mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k}$, then $\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3$.

• Applications of Scalar Product:

  • Finding the angle between vectors: $\cos(\theta) = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|}$.

  • Projection (Resolving one vector along another): The component of $\mathbf{a}$ in the direction of $\mathbf{n}$ is given by $\mathbf{a} \cdot \hat{\mathbf{n}}$.

  • Work Done ($W$): The work done by force $\mathbf{F}$ over displacement $\mathbf{r}$ is $W = \mathbf{F} \cdot \mathbf{r}$.

Electrostatics and Vector Fields

• Coulomb Force: The force between charges $q_1$ and $q_2$ is $\mathbf{F} = K \frac{q_1 q_2}{r^2} \hat{\mathbf{r}}$, where $r$ is distance and $K = \frac{1}{4 \pi \epsilon_0}$. $\epsilon_0 = 8.854 \times 10^{-12}\,F\,m^{-1}$.

• Electric Field ($E$): The total field at a point $G$ due to multiple point charges $q_i$ at positions $P_i$ is: $\mathbf{E} = \frac{1}{4 \pi \epsilon_0} \sum \frac{q_i}{|\mathbf{r}_i|^3} \mathbf{r}_i$, where $\mathbf{r}_i$ is the vector from $P_i$ to $G$.

• Field Example: For charges at corners of a cube ($1\,m$ side), the field at the center is zero due to symmetry. At the face center, the magnitude is calculated as $|\mathbf{E}| \approx 19.57\,V\,m^{-1}$. At the edge center, $|\mathbf{E}| \approx 25.72\,V\,m^{-1}$.

• Work in Electric Fields: Differential work in moving charge $q$ through displacement $d\mathbf{S}$ in field $\mathbf{E}$ is $W = -q\mathbf{E} \cdot d\mathbf{S}$.

The Vector Product (Cross Product)

• Definition: $\mathbf{a} \times \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \sin(\theta) \hat{\mathbf{e}}$, where $\hat{\mathbf{e}}$ is a unit vector perpendicular to the plane containing $\mathbf{a}$ and $\mathbf{b}$. The direction of $\hat{\mathbf{e}}$ is determined by the right-handed screw rule (turning from $\mathbf{a}$ to $\mathbf{b}$).

• Key Rules:

  • Anti-commutative: $\mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a})$.

  • Parallel vectors: $\mathbf{a} \times \mathbf{b} = \mathbf{0}$ if $\mathbf{a} || \mathbf{b}$.

  • Basis vectors: $\mathbf{i} \times \mathbf{i} = \mathbf{0}$, $\mathbf{i} \times \mathbf{j} = \mathbf{k}$, $\mathbf{j} \times \mathbf{k} = \mathbf{i}$, $\mathbf{k} \times \mathbf{i} = \mathbf{j}$. Reversed cycles yield negative unit vectors (e.g., $\mathbf{j} \times \mathbf{i} = -\mathbf{k}$).

• Determinant Formula: $\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} = \mathbf{i}(a_2 b_3 - a_3 b_2) - \mathbf{j}(a_1 b_3 - a_3 b_1) + \mathbf{k}(a_1 b_2 - a_2 b_1)$.

• Geometric Applications:

  • Area of a triangle with vertices $A, B, C$: $A_T = \frac{1}{2} |\vec{AB} \times \vec{AC}|$.

  • Moment (Torque) of a force: $\mathbf{M}_O = \mathbf{r} \times \mathbf{F}$, where $\mathbf{r}$ is the position vector from point $O$ to any point on the line of action of force $\mathbf{F}$.

Analytical Geometry: Lines and Planes

• Direction Ratios and Cosines:

  • For $\mathbf{r} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k}$, the direction ratios are $a:b:c$.

  • The direction cosines are $l = \frac{a}{|\mathbf{r}|}, m = \frac{b}{|\mathbf{r}|}, n = \frac{c}{|\mathbf{r}|}$, representing cosines of angles between the vector and axes. Note: $l^2 + m^2 + n^2 = 1$.

• Vector Equation of a Line: The equation of a line through points $A$ and $B$ with position vectors $\mathbf{a}$ and $\mathbf{b}$ is $\mathbf{r} = \mathbf{a} + t(\mathbf{b} - \mathbf{a})$, where $t$ is a scalar parameter.

• Cartesian Equation of a Line: $\frac{x - a_1}{b_1 - a_1} = \frac{y - a_2}{b_2 - a_2} = \frac{z - a_3}{b_3 - a_3}$.

• Vector Equation of a Plane:

  • A plane perpendicular to vector $\mathbf{n}$ through point with position vector $\mathbf{a}$: $(\mathbf{r} - \mathbf{a}) \cdot \mathbf{n} = 0$ or $\mathbf{r} \cdot \mathbf{n} = \mathbf{a} \cdot \mathbf{n}$.

  • Distance form: If $\hat{\mathbf{n}}$ is a unit normal and $d$ is the perpendicular distance from the origin, $\mathbf{r} \cdot \hat{\mathbf{n}} = d$.

• Example Plane Calculation: A plane through $(7, 3, -5)$ with normal $\mathbf{n} = (1, 1, 1)$ has the equation $\mathbf{r} \cdot \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} = 5$. Perpendicular distance from origin $d = \frac{5}{\sqrt{3}}$.