கணிதம் உங்களுக்காக - 19 - கான்ஸெப்ட்கள்

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

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

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

4. Direction cosines of a vector: cosα = A₁ / |A|, cosβ = A₂ / |A|, cosγ = A₃ / |A|

5. Volume of a truncated pyramid: V = (1/3)h(A₁ + A₂ + √(A₁A₂)) (where h is the height, A₁ is the area of the top base, and A₂ is the area of the bottom base)

6. Surface area of a truncated pyramid: SA = A₁ + A₂ + ½Pℓ (where P is the perimeter of the base and ℓ is the slant height)

7. Volume of a regular tetrahedron: V = (√2 / 12) a³ (where a is the side length)

8. Surface area of a regular tetrahedron: SA = √3 a² (where a is the side length)

9. Volume of an octahedron: V = (√2 / 3) a³ (where a is the side length)

10. Surface area of an octahedron: SA = 2√3 a² (where a is the side length)

11. Triple Angle Formulas:

    - sin(3A) = 3 sinA - 4 sin³A

    - cos(3A) = 4 cos³A - 3 cosA

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

12. 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)]

13. Distance from a point to a line in 3D: d = |Ax₁ + By₁ + Cz₁ + D| / √(A² + B² + C²)

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

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

16. Standard form of a parabola (vertical): y - k = a(x - h)² (where (h, k) is the vertex)

17. Standard form of a parabola (horizontal): x - h = a(y - k)² (where (h, k) is the vertex)

18. Standard form of an ellipse (centered at (h, k)): (x - h)² / a² + (y - k)² / b² = 1

19. Standard form of a hyperbola (horizontal): (x - h)² / a² - (y - k)² / b² = 1

20. Standard form of a hyperbola (vertical): (y - k)² / a² - (x - h)² / b² = 1

21. Rotation by θ degrees around a point (h, k): (x', y') = ((x - h) cosθ - (y - k) sinθ + h, (x - h) sinθ + (y - k) cosθ + k)

22. Dilation with center (h, k) and scale factor k: (x', y') = (h + k(x - h), k + k(y - k))

23. Translation: (x, y) → (x + a, y + b)

24. Reflection over the x-axis: (x, y) → (x, -y)

25. Reflection over the y-axis: (x, y) → (-x, y)

26. Reflection over the line y = x: (x, y) → (y, x)

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

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

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

30. Segment area: A = ½r²(θ - sinθ)

31. 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)

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

33. 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)