This chapter introduces the fundamental concepts of vectors and their operations, which are crucial in mathematics, physics, and engineering.
Vector Algebra – Formula & Equation Sheet
Essential formulas and equations from Mathematics Part - II, tailored for Class 12 in Mathematics.
This one-pager compiles key formulas and equations from the Vector Algebra chapter of Mathematics Part - II. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
Magnitude of a vector: |A| = √(x² + y² + z²)
Where A = (x, y, z) is a vector in 3D space. This formula calculates the length or magnitude of vector A.
Unit vector: Â = A / |A|
 represents the unit vector in the direction of vector A. It helps to express the direction of the vector without magnitude.
Position vector: r = xi + yj + zk
Where (x, y, z) are the coordinates of point P in 3D space, and i, j, k are unit vectors along the x, y, z axes, respectively.
Addition of vectors: R = A + B
If A and B are vectors, R is the resultant vector obtained by applying the triangle or parallelogram law.
Subtraction of vectors: R = A - B
This can be interpreted as R = A + (-B), where -B is the vector B in the opposite direction.
Dot product: A · B = |A||B|cosθ
Where θ is the angle between vectors A and B. This gives a scalar result and is useful in calculating angles.
Cross product: A × B = |A||B|sinθ n̂
Where n̂ is the unit vector perpendicular to the plane formed by A and B. The result is a vector.
Area of triangle: Area = 1/2 |A × B|
The area of the triangle formed by vectors A and B is half the magnitude of their cross product.
Direction cosines: l = cos(α), m = cos(β), n = cos(γ)
Where α, β, γ are the angles made by the vector with the x, y, and z axes, respectively.
Projection of vector A on vector B: proj_B(A) = (A · B / |B|²)B
This formula gives the component of vector A in the direction of vector B.
Equations
A + B = R
Vector addition where R is the resultant vector from vectors A and B.
A - B = R
Vector subtraction gives a resultant vector R arising from A and B.
A · B = |A||B|cosθ
The relation between dot product and angles provides insight into vector orientation.
A × B = |A||B|sinθ n̂
This equation defines the vector product, highlighting its geometric significance.
|A| = √(a² + b² + c²)
Used to find the magnitude of a vector given its components a, b, and c.
R = A + B + C
Resultant vector R from the addition of three vectors A, B, and C in a triangle.
l² + m² + n² = 1
The sum of the squares of the direction cosines equals 1 for any vector.
r = (x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2
Midpoint formula of vector joining points (x₁, y₁, z₁) and (x₂, y₂, z₂).
θ = cos⁻¹((A · B) / (|A||B|))
This equation computes the angle θ between two vectors using inverse cosine.
Area of parallelogram = |A × B|
The magnitude of the cross product of A and B gives the area of the parallelogram formed by them.
This chapter covers the concept of integrals, including indefinite and definite integrals, crucial for calculating areas under curves and solving practical problems in various fields.
Start chapterThis chapter explores how to use integrals to find areas under curves, between lines, and enclosed by shapes like circles and parabolas. Understanding these applications is crucial for solving real-world problems.
Start chapterThis chapter introduces differential equations, including their types and applications across various scientific fields.
Start chapterThis chapter focuses on the concepts and methods related to three-dimensional geometry, essential for understanding spatial relationships in mathematics.
Start chapterThis chapter focuses on linear programming, a method used to optimize certain objectives within given constraints, which is applicable in various fields like economics and management.
Start chapterThis chapter introduces the fundamental concepts of probability, including conditional probability and its applications which are essential for understanding uncertainty in random experiments.
Start chapter
