கணிதம் உங்களுக்காக - 25 - கான்ஸெப்ட்கள்

 

1. Equation of a line in space: (x - x₁) / a = (y - y₁) / b = (z - z₁) / c (where (x₁, y₁, z₁) is a point on the line and (a, b, c) is the direction vector)

2. Distance between two points in 3D: d = √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²)

3. Midpoint of a line segment in 3D: M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2, (z₁ + z₂) / 2)

4. Equation of a sphere (centered at the origin): x² + y² + z² = r²

5. Equation of a sphere (centered at (h, k, l)): (x - h)² + (y - k)² + (z - l)² = r²

6. Equation of a plane: Ax + By + Cz + D = 0

7. Distance from a point to a plane: d = |Ax₁ + By₁ + Cz₁ + D| / √(A² + B² + C²)

8. Volume of a regular polyhedron: V = (1/6) a³√(2/3)[5n(3 + 2√5)] (where a is the side length and n is the number of faces)

9. Surface area of a regular polyhedron: SA = n a²√3 (where a is the side length and n is the number of faces)

10. Volume of a truncated cube: V = (a³ / 3) (where a is the side length of the original cube and b is the side length of the truncated portion)

11. Surface area of a truncated cube: SA = 3a² + 2√3a² (where a is the side length of the original cube)

12. Volume of a cylindrical shell: V = 2πRh (R - r) (where R is the outer radius, r is the inner radius, and h is the height)

13. Surface area of a cylindrical shell: SA = 2πh(R + r) (where R is the outer radius, r is the inner radius, and h is the height)

14. Rotation around the origin by θ degrees: (x', y') = (x cosθ - y sinθ, x sinθ + y cosθ)

15. Reflection over a line y = mx + c: (x', y') = ((x(1 - m²) + 2my - 2mc) / (1 + m²), (y(1 - m²) + 2mx + 2c) / (1 + m²))

16. Dilation with respect to the origin: (x, y) → (kx, ky) (where k is the scale factor)

17. Horizontal shear transformation: (x', y') = (x + ky, y)

18. Vertical shear transformation: (x', y') = (x, y + kx)

19. Length of an arc: L = rθ (where r is the radius and θ is the central angle in radians)

20. Area of a sector: A = ½r²θ (where r is the radius and θ is the central angle in radians)

21. Segment area: A = ½r²(θ - sinθ)

22. Length of a chord: c = 2r sin(θ/2) (where r is the radius and θ is the central angle in radians)

23. Power of a Point Theorem: If a point P lies outside a circle and two secant lines PA and PB intersect the circle at A and B, then PA × PB = PC × PD (where C and D are points of intersection)

24. Segment lengths in a circle: If two chords AB and CD intersect at point P inside the circle, then PA × PB = PC × PD.

25. Length of a tangent segment: If a tangent from an external point P touches the circle at T, then PT² = PA × PB (where A and B are points of intersection of a secant through P)

26. Product-to-Sum Formulas:

    - sin(A) sin(B) = ½ [cos(A - B) - cos(A + B)]

    - cos(A) cos(B) = ½ [cos(A + B) + cos(A - B)]

    - sin(A) cos(B) = ½ [sin(A + B) + sin(A - B)]

27. Sum-to-Product Formulas:

    - sin(A) ± sin(B) = 2 sin((A ± B) / 2) cos((A ∓ B) / 2)

    - cos(A) + cos(B) = 2 cos((A + B) / 2) cos((A - B) / 2)

    - cos(A) - cos(B) = -2 sin((A + B) / 2) sin((A - B) / 2)

28. Half-Angle Formulas:

    - sin²(A/2) = (1 - cosA) / 2

    - cos²(A/2) = (1 + cosA) / 2

    - tan²(A/2) = (1 - cosA) / (1 + cosA)

29. Double Angle Formulas:

    - sin(2A) = 2 sinA cosA

    - cos(2A) = cos²A - sin²A = 2 cos²A - 1 = 1 - 2 sin²A

    - tan(2A) = 2 tanA / (1 - tan²A)

30. Triple Angle Formulas:

    - sin(3A) = 3 sinA - 4 sin³A

    - cos(3A) = 4 cos³A - 3 cosA

    - tan(3A) = (3 tanA - tan³A) / (1 - 3 tan²A)

31. Dot product: A · B = A₁B₁ + A₂B₂ + A₃B₃

32. Vector cross product: A × B = (A₂B₃ - A₃B₂)i + (A₃B₁ - A₁B₃)j + (A₁B₂ - A₂B₁)k

33. Magnitude of a vector in 3D: |A| = √(A₁² + A₂² + A₃²)

34. Scalar projection of vector A onto vector B: proj_B(A) = (A · B) / |B|

35. Vector projection of vector A onto vector B: Proj_B(A) = ((A · B) / |B|²) B

36. Angle between two vectors: cos(θ) = (A · B) / (|A||B|)

37. Distance between two skew lines: d = |(A₁ - A₂) · (B₁ × B₂)| / |B₁ × B₂| (where A₁ and A₂ are points on the lines, and B₁ and B₂ are direction vectors)

38. Volume of a parallelepiped: V = |A · (B × C)| (where A, B, and C are vectors)

39. Scalar triple product: V = A · (B × C) (where A, B, and C are vectors)

40. Vector triple product: A × (B × C) = (A · C)B - (A · B)C