Algebra: Basics & Vector Algebra

Algebra is a branch of mathematics that involves using symbols and letters to stand in for numbers and express mathematical relationships. It bridges arithmetic and more advanced mathematics, enabling us to generalize formulas, solve equations, and model real-world problems. From simple expressions to abstract theories, algebra serves as a foundation in nearly every field of mathematics and science.

1.0What is Algebra?

In simple terms, algebra allows us to form equations using variables (like x, y, and z) and constants. These variables can represent unknowns or varying values.

Key Components of Algebra:

• Variables: Symbols that represent unknown values.
• Constants: Known, fixed values.
• Expressions: Mathematical phrases involving variables, numbers, and operations.
• Equations: Statements showing the equality of two expressions.
• Functions: Relationships between input and output variables.

Also Read: The Fundamentals of Algebra

2.0Linear Algebra and Its Applications

Linear algebra is a special branch of algebra that deals with vectors, matrices, and linear transformations. It is crucial in both pure and applied mathematics, engineering, physics, computer science, and data science.

Key Topics in Linear Algebra:

• Matrix operations
• Determinants
• Vector spaces
• Eigenvalues and Eigenvectors
• Systems of linear equations

3.0Applications of Linear Algebra:

• Machine Learning: Algorithms like SVMs, PCA, and neural networks.
• Computer Graphics: Modeling and transforming images using matrices.
• Economics: Solving systems in input-output models.
• Engineering: Structural analysis, electrical circuits, and control systems.

4.0Class 12 Vector Algebra – Overview

Vector Algebra is a core part of Class 12 mathematics, especially for JEE aspirants and science students. It focuses on the mathematical study of vectors, which are quantities having both magnitude and direction.

Important Concepts in Class 12 Vector Algebra:

• Vector addition and subtraction
• Dot product (scalar product)
• Cross product (vector product)
• Section formula
• Direction cosines and direction ratios
• Applications in geometry and physics

5.0Why is Algebra Important?

• Builds logical thinking and problem-solving skills.
• Essential for science, technology, engineering, and finance.
• Forms the basis for calculus, statistics, and data analysis.
• Widely used in daily life – budgeting, coding, business modeling, and more.

6.0Solved Questions on Algebra 

Example 1: Solve for x: $2x^2-5x+3=0$

Solution:

Using the quadratic formula:

$x=\frac{-(-5)\pm \sqrt{(-5{)}^2-4(2)(3)}}{2(2)}.$

$x=\frac{5\pm \sqrt{25-24}}{4}=\frac{5\pm 1}{4}.$

$\Rightarrow x=\frac{6}{4}=\frac{3}{2},x=\frac{4}{4}=1.$

Answer: x = 1, $\frac{3}{2}$

Example 2: Simplify: $(x+y{)}^3-(x-y{)}^3$

Solution:

Using identity:

$a^3-b^3=(a-b)(a^2+ab+b^2).$

Let a = x + y, b = x - y

$[(x+y)-(x-y)]\cdot [(x+y{)}^2+(x+y)(x-y)+(x-y{)}^2]$

$=2y\cdot [(x^2+2xy+y^2)+(x^2-y^2)+(x^2-2xy+y^2)]$

$=2y\cdot [3x^2+2y^2]$

Answer: $=2y\cdot [3x^2+2y^2]$

Example 3: If $\vec{a}=\hat{i}+2\hat{j}+3\hat{k},\vec{b}=4\hat{i}-\hat{j}+2\hat{k}$ , find $\vec{a}\cdot \vec{b}$

Solution:

Dot product:

$\vec{a}\cdot \vec{b}=(1)(4)+(2)(-1)+(3)(2)$

$=4-2+6=8.$

Answer: 8

Example 4: If $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$, find the determinant of A.

Solution:

|A| = (1)(4) - (2)(3) = 4 - 6 = -2 

Answer: -2

Example 5: Find $\vec{a}\times \vec{b}$, where  $\vec{a}=\hat{i}+\hat{j},\vec{b}=\hat{j}+\hat{k}$

Solution:

$\hat{a}\times \hat{b}=\left|\begin{array}{ccc}\hat{ı}& \hat{ȷ}& \hat{k}\\ 1& 1& 0\\ 0& 1& 1\end{array}\right|$

$=\hat{ı}(1\cdot 1-0\cdot 1)-\hat{ȷ}(1\cdot 1-0\cdot 0)+\hat{k}(1\cdot 1-1\cdot 0)$

$=\hat{ı}(1)-\hat{ȷ}(1)+\hat{k}(1)$

$=\hat{ı}-\hat{ȷ}+\hat{k}$

Answer: $\hat{ı}-\hat{ȷ}+\hat{k}$

7.0Practice Questions on Algebra

1. Solve $x^3-6x^2+11x-6=0$
2. If a + b = 10 and ab = 24, find $a^2+b^2$.
3. Factorize $2x^2-7x+3$:
4. Find the unit vector in the direction of vector $\hat{a}=3\hat{ı}-4\hat{ȷ}$
5. Show that vectors $\hat{a}=2\hat{ı}-\hat{ȷ}$, $\hat{b}=4\hat{ı}-2\hat{ȷ}$ are collinear.
6. If matrix $A=\left[\begin{array}{cc}2& 3\\ 0& 1\end{array}\right]$, find $A^2$.
7. Find the inverse of the matrix $\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$, if it exists.