VECTORS

• Vector Arithmetic
    - Mathematical operations involving vectors, such as addition, subtraction, dot product, and cross product.
  

Vector Algebra

• Definition:
    - A vector is a quantity defined by its magnitude and direction.
    - Notation:
      - For a two-dimensional vector, the components are denoted as extbf{v} = egin{pmatrix} x_1 \ y_1 \ ext{and for a three-dimensional vector, it is denoted as } extbf{v} = egin{pmatrix} x_1 \ y_1 \ z_1 \.

• Given two vectors $extbf{u}$ and $extbf{v}$:
    - (i) Addition:
       extbf{u} + extbf{v} = egin{pmatrix} u_1 + v_1 \ u_2 + v_2 \ u_3 + v_3 \
    - (ii) Subtraction:
       extbf{u} - extbf{v} = egin{pmatrix} u_1 - v_1 \ u_2 - v_2 \ u_3 - v_3 \
    - (iii) If $k$ is a scalar, then:
      k extbf{u} = egin{pmatrix} k u_1 \ k u_2 \ k u_3 \

Properties of Vector Operations

• Addition of Vectors:
    - Commutative: $extbf{u} + extbf{v} = extbf{v} + extbf{u}$.

• Subtraction of Vectors:
    - Non-commutative: $extbf{u} - extbf{v} <br>eq extbf{v} - extbf{u}$.

Basic Unit Vectors

• When vectors are expressed in column form:
    - The basic unit vectors in the x-, y-, and z-directions are denoted as:
       extbf{i} = egin{pmatrix} 1 \ 0 \ 0 \, extbf{j} = egin{pmatrix} 0 \ 1 \ 0 \, extbf{k} = egin{pmatrix} 0 \ 0 \ 1 \.
    - A vector $extbf{v}$ can be represented as:
      $extbf{v} = x extbf{i} + y extbf{j} + z extbf{k}$.

Scalar (Dot) Product

• Definition:
    - The scalar or dot product of two vectors $extbf{u}$ and $extbf{v}$ is given by:
      $extbf{u} \bullet extbf{v} = | extbf{u}|| extbf{v}| ext{cos}( heta)$,
      where $heta$ is the angle between the two vectors.
  

• Properties:
    - (i) $extbf{u} \bullet extbf{u} = | extbf{u}|^2$.
    - (ii) $extbf{u} \bullet extbf{v} = 0 ext{ iff } extbf{u} ext{ and } extbf{v} ext{ are perpendicular}.$
    - (iii) Using components, $extbf{u} \bullet extbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3$.

• Compare:
    - Notice:
      $| extbf{u}| = extbf{u} \bullet extbf{u}$.

Solved Questions

• Question 1:
    - Vectors extbf{u} = egin{pmatrix} -2 \ 6 \ -3 \ and extbf{v} = egin{pmatrix} -2 \ -3 \ 6 \. Calculate the acute angle between them.

• Question 2:
    - Given points A, B, C with position vectors extbf{a} = egin{pmatrix} 2 \ 1 \ 2 \, extbf{b} = egin{pmatrix} -3 \ 2 \ 5 \, extbf{c} = egin{pmatrix} 4 \ 5 \ -2 \.
      - Find the position vector of D where ABCD forms a parallelogram.
      - Find the position vector for the intersection of diagonals AC and BD.
      - Calculate angle $ext{BAC}$.
  

Vector Product

• Definition:
    - The cross product $extbf{u} imes extbf{v}$ is defined as:
      $extbf{u} imes extbf{v} = | extbf{u}|| extbf{v}| ext{sin}( heta) extbf{n}$,
      where $extbf{n}$ is a unit vector perpendicular to both $extbf{u}$ and $extbf{v}$.

Properties of Cross Product

• (1) $extbf{u} imes extbf{v} = - extbf{v} imes extbf{u}$ (Anticommutative).

• (2) $extbf{u} imes extbf{v} = extbf{0}$ for parallel (angle 0° or 180°).

Area of a Triangle

• Formula:
      - The area of a triangle formed by vectors $extbf{u}$ and $extbf{v}$ can be computed using:
      $ext{Area} = rac{1}{2} | extbf{u} imes extbf{v}|$.

Integration

• Integration Overview:
    - Integration is defined as the reverse operation of differentiation. Given a function $f(x)$ and its derivative $f'(x)$, integrating $f'(x)$ returns $f(x)$ plus an arbitrary constant.

Types of Integrals

• Indefinite Integral:
    - Represents a family of functions with the general form $extstyle ext{∫} f(x) ext{d}x = F(x) + C$ where C is a constant.

• Definite Integral:
    - Gives the signed area under the curve of the function from point a to point b:
      $extstyle ext{∫}_{a}^{b} f(x) ext{d}x = F(b) - F(a)$.

Improper Integrals

• Definition:
    - An improper integral occurs when the limits of integration include infinity or the integrand approaches infinity within the limits.

• Evaluation:
    - For instance, to evaluate the improper integral, replace $extstyle ±∞$ with variable $n$ and find the limit of the integral as $n o ±∞$.

Integration Techniques

• Substitution:
    - A method to simplify integration processes by changing variables.

• Integration by Parts:
    - Based on the product rule of differentiation:
      $extstyle ext{∫} u ext{d}v = uv - extstyle ext{∫} v ext{d}u$.
  

Special Integrals and Areas

• Volume of Solids of Revolution:
    - When a function $y = f(x)$ is revolved around the x-axis, the volume of revolution can be calculated using the formula
      $V = extstyle ext{π} ext{∫}_{a}^{b} [f(x)]^2 ext{d}x$.

MATRICES

• Definition:
    - A matrix is a rectangular or square array of numbers represented as $m imes n$ (rows × columns).

• Matrix Algebra:
    - Operations include addition, subtraction, and multiplication of matrices.

Determinants

• Definition:
    - The determinant provides a scalar value that is a function of the entries of a square matrix, calculated using formulas appropriate for the size of the matrix.

Systems of Linear Equations

• Conversion to Matrix Form:
    - Linear equations can be arranged into the matrix form $AX = B$, where A is the coefficients matrix, X is the variable matrix, and B is the constant matrix.

• Cramer's Rule:
    - For a system of linear equations, it can be solved using the determinants of the matrices as follows:
      $X_i = rac{det(A_i)}{det(A)}$, where $A_i$ is derived from A by replacing the i-th column with the constants from the equations.

Practical Applications

• Areas of integration are useful in economics (area under demand curves), biology (population models), and physics (work done brings energy considerations).

• Matrix applications emerge in system modeling, computer graphics, and solving systems of equations in various scientific fields.

VECTORS - Vector Arithmetic

 - Vector arithmetic involves mathematical operations with vectors, which are quantities having both magnitude and direction. Key operations include addition, subtraction, the dot product, and the cross product, which are essential for various applications in physics and engineering.

Vector Algebra - Definition:

 - A vector is a mathematical representation of a quantity that has both size (magnitude) and direction.

 - Notation:

 - For a two-dimensional vector, the components are denoted as $\mathbf{v} = \begin{pmatrix} x<em>1 \ y</em>1 \end{pmatrix}$, and for a three-dimensional vector, it is represented as $\mathbf{v} = \begin{pmatrix} x<em>1 \ y</em>1 \ z<em>1 \end{pmatrix}$.  - Given two vectors $\mathbf{u}$ and $\mathbf{v}$:
 - (i) Addition:
 $\mathbf{u} + \mathbf{v} = \begin{pmatrix} u</em>1 + v<em>1 \ u</em>2 + v<em>2 \ u</em>3 + v<em>3 \end{pmatrix}$
 - (ii) Subtraction:
 $\mathbf{u} - \mathbf{v} = \begin{pmatrix} u</em>1 - v<em>1 \ u</em>2 - v<em>2 \ u</em>3 - v<em>3 \end{pmatrix}$
 - (iii) If $k$ is a scalar, then:
 $k \mathbf{u} = \begin{pmatrix} k u</em>1 \ k u<em>2 \ k u</em>3 \end{pmatrix}$

Properties of Vector Operations - Addition of Vectors:

 - Commutative: The order in which vectors are added does not affect the result, meaning $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$.
 - Subtraction of Vectors:

 - Non-commutative: The order of subtraction matters, hence $\mathbf{u} - \mathbf{v} \neq \mathbf{v} - \mathbf{u}$.

Basic Unit Vectors - When vectors are expressed in column form:

 - The basic unit vectors in the x-, y-, and z-directions are represented as:

 $\mathbf{i} = \begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix}$, $\mathbf{j} = \begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix}$, $\mathbf{k} = \begin{pmatrix} 0 \ 0 \ 1 \end{pmatrix}$.

 - Any vector $\mathbf{v}$ in three-dimensional space can also be expressed as:

 $\mathbf{v} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k}$, indicating that it consists of contributions along each of the three axes.

Scalar (Dot) Product - Definition:

 - The scalar or dot product of two vectors $\mathbf{u}$ and $\mathbf{v}$ is expressed mathematically as:

 $\mathbf{u} \bullet \mathbf{v} = | \mathbf{u} | | \mathbf{v} | \cos(\theta)$,

 where $\theta$ represents the angle between the two vectors. This operation returns a scalar value, which helps quantify how much one vector extends in the direction of another.
 - Properties:

 - (i) The dot product of a vector with itself gives the square of its magnitude: $\mathbf{u} \bullet \mathbf{u} = | \mathbf{u} |^2$.

 - (ii) The dot product equals zero if the vectors are perpendicular (orthogonal): $\mathbf{u} \bullet \mathbf{v} = 0 \text{ if and only if } \mathbf{u} \text{ and } \mathbf{v} \text{ are perpendicular}.$

 - (iii) Using the component representation, the dot product is calculated as: $\mathbf{u} \bullet \mathbf{v} = u<em>1 v</em>1 + u<em>2 v</em>2 + u<em>3 v</em>3$.

• Compare:
  
   - Notice:
  
   $| \mathbf{u}| = \mathbf{u} \bullet \mathbf{u}$, providing a geometric interpretation of the dot product.

Solved Questions - Question 1:

 - Given vectors $\mathbf{u} = \begin{pmatrix} -2 \ 6 \ -3 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} -2 \ -3 \ 6 \end{pmatrix}$, calculate the acute angle between them using the dot product formula.

 - Question 2:

 - Given points A, B, and C with position vectors $\mathbf{a} = \begin{pmatrix} 2 \ 1 \ 2 \end{pmatrix}$, $\mathbf{b} = \begin{pmatrix} -3 \ 2 \ 5 \end{pmatrix}$, and $\mathbf{c} = \begin{pmatrix} 4 \ 5 \ -2 \end{pmatrix}$:

 - Find the position vector of point D such that ABCD forms a parallelogram.

 - Calculate the position vector for the intersection of diagonals AC and BD.

 - Compute the angle $\angle BAC$.

Vector Product - Definition:

 - The cross product, denoted $\mathbf{u} \times \mathbf{v}$, is defined mathematically as:

 $\mathbf{u} \times \mathbf{v} = | \mathbf{u} | | \mathbf{v} | \sin(\theta) \mathbf{n}$,

 where $\mathbf{n}$ is a unit vector that is perpendicular to both $\mathbf{u}$ and $\mathbf{v}$, thus giving a vector result.

Properties of Cross Product - (1) The cross product is anticommutative: $\mathbf{u} \times \mathbf{v} = - \mathbf{v} \times \mathbf{u}$.

• (2) The cross product yields the zero vector if the vectors are parallel (angle of either 0° or 180°).

Area of a Triangle - Formula:

 - The area of a triangle formed by vectors $\mathbf{u}$ and $\mathbf{v}$ can be derived from the cross product as:

 $\text{Area} = \frac{1}{2} | \mathbf{u} \times \mathbf{v} |$, providing a method to calculate areas geometrically.

Integration - Integration Overview:

 - Integration is defined as the inverse operation of differentiation. If we have a function $f(x)$ and its derivative $f'(x)$, performing an integration on $f'(x)$ will return $f(x)$ plus a constant of integration, often denoted as C.

Types of Integrals - Indefinite Integral:

 - Represents a family of functions, typically written in the form $\int f(x) \,dx = F(x) + C$, showing that there are multiple antiderivatives differing by a constant C.

• Definite Integral:
  
   - Represents the signed area under the curve of the function $f(x)$ from point a to point b, calculated as:
  
   $\int_{a}^{b} f(x) \,dx = F(b) - F(a),$ where $F$ is the antiderivative of $f$.

Improper Integrals - Definition:

 - An improper integral occurs when either the limits of integration involve infinity or the integrand approaches infinity within the limits.

 - Evaluation:

 - To evaluate an improper integral, replace the bounds involving infinity with a variable, say $n$, and find the limit of the integral as $n$ approaches the specified limit (either $\pm \infty$).

Integration Techniques - Substitution:

 - A method that simplifies the integration process by changing the variables to make the integral more manageable.

• Integration by Parts:
  
   - This technique is based on the product rule from differentiation:
   $\int u \,dv = uv - \int v \,du$, simplifying complex integrals through structured choices for u and v.

Special Integrals and Areas - Volume of Solids of Revolution:

 - When a given function $y = f(x)$ is revolved around the x-axis, the volume of the resulting solid can be computed using the formula:
 $V = \pi \int_{a}^{b} [f(x)]^2 \,dx$, incorporating the disk method for calculation.

MATRICES - Definition:

 - A matrix is a rectangular or square array of numbers, usually denoted by its dimensions, for example, $m \times n$ (rows × columns), facilitating systematic representation of data.

• Matrix Algebra:
  
   - Operations involving matrices include addition, subtraction, and multiplication, which follow specific rules and properties that allow for manipulation of linear equations efficiently.

Determinants - Definition:

 - The determinant provides a scalar value that relates to the properties of a square matrix, calculated using methods tailored to the matrix size, which aids in solving systems of linear equations and assessing invertibility.

Systems of Linear Equations - Conversion to Matrix Form:

 - A system of linear equations can be arranged in a matrix form as $AX = B$, where A contains the coefficients of the variables, X is the matrix of variables, and B holds the constant terms.

• Cramer's Rule:
  
   - A method for solving systems of equations leveraging determinants, given by $X<em>i = \frac{det(A</em>i)}{det(A)}$, whereby $A_i$ is formed by replacing the i-th column of A with the constants from the equations.

Practical Applications - The areas of integration are crucial in fields like economics (to analyze areas under curves representing demand), biology (population growth models), and physics (calculating work done, integrating force over a distance). - Matrix applications span across areas like system modeling, computer graphics