1. Scalar projection of vector A onto vector B: proj_B(A) = (A · B) / |B|
2. Vector projection of vector A onto vector B: Proj_B(A) = ((A · B) / |B|²) B
3. Angle between two vectors: cos(╬╕) = (A · B) / (|A||B|)
4. Magnitude of a vector: |A| = √(A₁² + A₂² + A₃²)
5. Cross product of two vectors: A × B = |A||B|sin(╬╕)n (where n is the unit vector perpendicular to the plane containing A and B)
6. Equation of a line in vector form: r = r₀ + t(v) (where r is the position vector, r₀ is a point on the line, v is the direction vector, and t is a scalar)
7. Equation of a plane in vector form: r · n = d (where r is the position vector, n is the normal vector, and d is the distance from the origin)
8. Distance between two points in space: d = √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²)
9. Midpoint between two points in space: M = ((x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2)
10. Equation of a sphere: (x - h)² + (y - k)² + (z - l)² = r² (where (h, k, l) is the center and r is the radius)
11. Volume of a parallelepiped: V = |a · (b × c)| (where a, b, and c are vectors representing the edges)
12. Equation of an ellipse (centered at (h, k)): ((x - h)² / a²) + ((y - k)² / b²) = 1 (where a is the semi-major axis and b is the semi-minor axis)
13. Equation of a hyperbola (centered at (h, k)): ((x - h)² / a²) - ((y - k)² / b²) = 1 (horizontal) or ((y - k)² / a²) - ((x - h)² / b²) = 1 (vertical)
14. Asymptotes of a hyperbola (horizontal): y - k = ± (b / a)(x - h)
15. Asymptotes of a hyperbola (vertical): y - k = ± (a / b)(x - h)
16. Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)
17. Law of Cosines: c² = a² + b² - 2ab cos(C)
18. Law of Tangents: (a - b) / (a + b) = tan[(A - B)/2] / tan[(A + B)/2]
19. Area of a triangle using trigonometry: A = ½ab sin(C)
20. Number of diagonals in a polygon: n(n - 3) / 2 (where n is the number of sides)
21. Measure of each interior angle of a regular polygon: (n - 2) × 180° / n
22. Measure of each exterior angle of a regular polygon: 360° / n
23. Perimeter of a regular polygon: P = n × s (where n is the number of sides and s is the side length)
24. Sum of interior angles of a polygon: (n - 2) × 180° (where n is the number of sides)
25. Sum of exterior angles of a polygon: 360°
26. Area of a regular polygon: A = ½ × Perimeter × Apothem (where Perimeter = n × s, n is the number of sides, s is the side length, and the Apothem is the perpendicular distance from the center to a side)
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