Mathematical Tools

1. Unitary Method

• Unitary method : first find the value of a single unit, then multiply (or divide) to get the value for the required number of units.

e.g. ₹500 → 20 mangoes ⇒ ₹1 → 20/500 ⇒ ₹300 → 20500 × 300 = 12 mangoes.

• Uniform rod (mass m, length l) : mass of a part of length x = ml x.
• Uniformly charged ring (total charge Q) : charge on an arc of angle θ = Q360° × θ.

2. Uniform (Linear) Change with Time

• A quantity changes uniformly (linearly) if it changes by the same amount in every equal interval.

a → change per second (rate of change) = Q₂ − Q₁t₂ − t₁ ;  b → initial value (at t = 0)

• Same form as a straight line y = mx + c (m = slope, c = intercept).
• e.g. speed 10 m/s at t = 0, increasing by 2 m/s every second ⇒ v = 2t + 10 ; at t = 9 s, v = 9 × 2 + 10 = 28 m/s.

Graph :

• Straight line ⇒ slope y₂ − y₁x₂ − x₁ is constant ⇒ equal Δx gives equal Δy ⇒ change is uniform.
• Curve ⇒ equal Δx gives unequal Δy ⇒ change is not uniform (Q = at + b cannot be used).

3. Uniform Change along a Length or Scale

• The same rule works when a quantity changes uniformly with position x (or with the number of divisions of a scale) :

a = change in Qchange in x (change per unit length) ;  b = value at x = 0

• Rod temperature 25 °C (end A) → 175 °C (end B), length 15 m : rate = 175 − 2515 = 10 °C/m ⇒ T = 10x + 25 ; at x = 3 m, T = 55 °C.
• Uniform resistor (length l), ends at 30 V and 100 V : V = 70lx + 30 ; mid-point (x = l/2) : V = 35 + 30 = 65 V.
• Meter scale : ammeter, full scale 20 A over 100 divisions ⇒ 20100 = 15 A/div ; 45 divisions ⇒ i = 15 × 45 + 0 = 9 A.

4. Density & Conservation of Mass

• Unit kg/m³. Density depends only on the material, not on shape or size ⇒ different objects made of the same material have the same density.
• Volumes : sphere V = 43πR³ ; cube V = a³.
• Same material ⇒ m ∝ V : m₂m₁ = V₂V₁. e.g. sphere of radius R has mass 10 kg ⇒ radius 2R (same material) has 2³ × 10 = 80 kg.

Conservation of mass :

• On melting, freezing, merging or reshaping, mass is conserved ; volume is not conserved in general (water 1 g/cm³ → ice 0.9 g/cm³ : same mass, larger volume).
• Same material (density unchanged) ⇒ volume is also unchanged.
• n equal drops merge into one : n · 43πr³ = 43πR³ ⇒ R = n^1/3 r (64 drops of 7 mm ⇒ R = 4 × 7 = 28 mm).
• Sphere melted into a cube : ρ43πR³ = ρa³ ⇒ a = (43π)^1/3 R.
• Body with a cavity : m_remaining = m_full − m_cavity (cavity treated as a body of the same material).

5. Scaling of Objects (Replica Rule)

• If every dimension of an object is made N times with no change in shape (a replica), for any shape :

Sphere : V₂ = 43π(NR)³ = N³V₁ ;  surface area A₂ = 4π(NR)² = N²A₁

• 2-D object (photographic enlargement) : every length × N ⇒ area A → N²A.
• Mass m = ρV : same density ⇒ m × N³ (e.g. heart twice as big, 500 g ⇒ 8 × 500 = 4000 g).
• Different density : m₂m₁ = ρ₂ρ₁ · N³.

6. Useful Results from Basic Maths

• Linear variation : y = ax + b ⇒ a = rate of change (slope), b = initial value (value at x = 0).
• e.g. speed in a pipe of radius R varies uniformly from 3v₀ (axis) to v₀ (wall) : rate = (v₀ − 3v₀)/R ⇒ v = 3v₀ − 2v₀Rr (check : r = 0 ⇒ 3v₀ ; r = R ⇒ v₀).

• Size made 1/5 at the same mass : V₂ = V₁/125 ⇒ ρ₂ = 125ρ₁ (ρV = constant).
• Mass = ρ × volume ; sphere V = 43πR³ ⇒ M ∝ ρR³ (radius 2R, density 3ρ ⇒ 3 × 8 = 24 times the mass).

e.g. core of density 2ρ up to R/2, outer part ρ from R/2 to R : M_outer/M_core = ρ(1 − ⅛)/(2ρ · ⅛) = 7/2.

7. Trigonometric Ratios

• One ratio given ⇒ draw the triangle, find the missing side by Pythagoras, then read the others. e.g. sin x = 2/3 ⇒ B = √(9 − 4) = √5 ⇒ cos x = √5/3, tan x = 2/√5.

8. Values at Standard Angles

37° and 53° come from the 3 : 4 : 5 triangle (sin37° = cos53° = 3/5 ; cos37° = sin53° = 4/5).

9. Units of Angle – Degree & Radian

• Radian is the SI unit (the "real" unit). Degree may be used only inside sin, cos, tan : sin60° = sin(π/3).
• If an angle is multiplied by any physical quantity, it must be in radian.

60° = π/3 ; 30° = π/6 ; 45° = π/4 (multiply degrees by π/180).

10. Arc Length & Sector Area

• θ must be in radian. e.g. θ = 45° : l = R(45° × π/180) = πR/4 (not R × 45) ; A = ½R²(π/4) = πR²/8.

11. Trigonometric Identities

Negative angle :

e.g. sin(−30°) = −½ ; cos(−30°) = cos30° = √3/2

Compound angles :

15° and 75° (use 45° ∓ 30°) :

e.g. sin15° = sin45°cos30° − cos45°sin30° = 1√2·√32 − 1√2·12 ; tan15° can also be found as sin15°/cos15°.

12. Double Angle & Half Angle Results

• Write cosA = cos(2 · A/2) and use the two forms of cos2A :

13. Angles of an N-sided Polygon

• Sum of all exterior angles of any polygon = 360° = 2π : θ₁ + θ₂ + … + θ_N = 2π.
• Sum of all interior angles = (N − 2)180° = (N − 2)π.

Interior φ = π − θ at each corner ⇒ Σφ = (π − θ₁) + (π − θ₂) + … (N terms) = Nπ − 2π = (N − 2)π

• Sum of the angles made by all sides at the centre = 360° ⇒ in a regular polygon each side makes 2π/N at the centre.
• e.g. 12-sided polygon : interior sum = (12 − 2)π = 10π.

14. Regular Polygon – Distances from the Centre

• Regular polygon : all sides equal (a), all interior angles equal.

• Join C to the ends of one side (angle 2π/N) and drop ⊥ r₁ on it : right triangle with angle π/N at C, opposite side a/2.

tan(π/N) = a/2r₁  ;  sin(π/N) = a/2r₂

Pentagon : each side makes 72° at the centre (half-angle 36°). Hexagon : 60° at the centre, so it splits into 6 equilateral triangles ⇒ r₂ = a.

15. Binomial Approximation

• x and n may be positive, negative or fractions. x ≪ 1 means roughly x < 0.1 (1/10).
• If required (e.g. first-order terms cancel), keep the next term :

• Method : write the number as (1 ± small) and use 1 + nx with the sign of x. e.g. (0.99)⁻⁷ = (1 + (−0.01))⁻⁷ ≈ 1 + (−7)(−0.01) = 1.07.
• It is approximate : (1.1)² = 1.21 exactly, but 1 + 2(0.1) = 1.2.
• More : (1.02)^1/4 ≈ 1 + 0.02/4 = 1.005 ; (1.02)⁻⁵ ≈ 0.9 ; (0.97)⁻⁴ ≈ 1.12.

16. Binomial Approximation for (a + x)ⁿ

• If x ≪ a then x/a ≪ 1 ; take aⁿ common :

(a + x)ⁿ = [a(1 + x/a)]ⁿ = aⁿ(1 + x/a)ⁿ

Powers : 2^a·2^b = 2^a+b ; (2^a)^b = 2^ab ; 2^a/2^b = 2^a−b ; e.g. (10²)^1/2 = 10.

• (101)⁷ = (100)⁷(1 + 1/100)⁷ ≈ 10¹⁴ × 1.07 = 107 × 10¹².
• (102)⁴ ≈ 10⁸(1 + 4 × 2/100) = 108 × 10⁶ ;  (102)^1/2 ≈ 10(1 + ½ × 2/100) = 10.1.
• (31)⁴ − (30)⁴ = 30⁴[(1 + 1/30)⁴ − 1] ≈ 30⁴ × 4/30 = 4 × 30³ = 108 × 10³.

Standard results (x ≪ a) :

• √(a² − x²) − a = a(1 − x²a²)^1/2 − a ≈ a(1 − x²2a²) − a = −x²2a
• KQx(a² + x²)^3/2 = KQx(a²)^3/2(1 + x²/a²)^3/2 ≈ KQxa³(1 + 3x²/2a²)

17. Small Angle (Trigonometric) Approximation

• Valid when the angle is very small : x < 10° in degree ; x ≪ 1 in radian.
• x must be in radian in sin x ≈ x ≈ tan x.
• If required : cos x ≈ 1 − x²2.

18. Sphere & Spherical Cap

• Cap of height h cut from a sphere of radius R ; θ = half-angle of the cap at the centre C (h = R − R cosθ).

• Hemisphere (h = R) : A = 2πR², V = πR²3(3R − R) = 23πR³.
• Satellite at altitude R above the earth (radius R) : the visible cap is bounded by tangent lines from the satellite ⇒ cosθ = R2R = ½ (θ = 60°, h = R/2) ⇒ area covered = 2πR²(1 − ½) = πR².

19. Straight Line : Slope & Intercept

• y changes uniformly (linearly) with x.
• m → slope = tanθ ; θ → angle of the line with the +ve x-axis (measured anticlockwise).
• c → y-intercept (where the line cuts the y-axis).

• Slope from two points (x₁, y₁) and (x₂, y₂) on the line :

• c > 0 : line cuts the y-axis above O ; c = 0 : line passes through the origin, y = mx ; c < 0 : cuts below O.

20. Sign of the Slope

• Parallel lines make the same θ ⇒ same slope.
• Falling line making acute angle φ with the x-axis : θ = 180° − φ ⇒ m = tan(180° − φ) = −tanφ.

e.g. θ = 120° (φ = 60°) : m = tan120° = −tan60° = −√3. A v–t graph rising at 60°, then flat, then falling at 60° : slopes √3, 0, −√3.

21. Comparing Slopes

• Positive slopes : larger θ (θ₂ > θ₁) ⇒ tanθ₂ > tanθ₁ ⇒ m₂ > m₁.
• Greater |m| ⇒ steeper (more vertical) line — for both positive and negative slopes.
• Negative slopes : compare by value and by magnitude separately.

e.g. falling lines at φ = 30° and 60° with the x-axis : m₁ = −tan30° = −1/√3, m₂ = −tan60° = −√3.
By value : −√3 < −1/√3 ⇒ m₂ < m₁ ; by magnitude : |m₂| = √3 > |m₁| = 1/√3 ⇒ line 2 is steeper.

22. Slope of the Tangent to a Curve

• Tangent to a 2-D (x–y) curve : the line which just touches the curve at a point.

v–t graph :

• Straight line through O : v increasing, slope constant ⇒ a constant.
• Curve bending upward : v increasing, slope increasing ⇒ a increasing.
• Curve bending over (flattening) : v increasing, slope decreasing ⇒ a decreasing.

Change of |slope| along a curve (moving forward in x) :

• Tangent becoming more vertical ⇒ |m| increases ; becoming more horizontal ⇒ |m| decreases.

23. Slope in a Momentum–Time Graph

24. Maxima & Minima

• At a local maximum or local minimum the tangent is horizontal ⇒ slope = 0.
• Value of y at a local maximum is higher than the values of y at nearby points on either side of it (lower for a local minimum).

• For the cubic y = ax³ + bx² + cx + d above : at x₁ there is a local maximum (dy/dx = 0, d²y/dx² < 0), but y at x₁ is not the greatest value of y — the curve goes higher further right.
• For a parabola y = ax² + bx + c opening upward, the local minimum at x₁ is also the least value of y.

Method :

• Put dy/dx = 0 → find x ; check the sign of d²y/dx² at that x ; substitute x in y for y_max / y_min.

e.g. y = x² − 8x + 10 : dy/dx = 2x − 8 = 0 ⇒ x = 4 ; d²y/dx² = 2 > 0 ⇒ minimum, y_min = 16 − 32 + 10 = −6.
y = 3x² − 12x + 100 : x = 2, d²y/dx² = 6 > 0 ⇒ y_min = 12 − 24 + 100 = 88.
y = 2x³ − 15x² + 36x + 12 : dy/dx = 6x² − 30x + 36 = 0 ⇒ (x − 2)(x − 3) = 0 ; d²y/dx² = 12x − 30 = −6 at x = 2 (maximum), +6 at x = 3 (minimum).

25. Maxima & Minima – Applications

• To find a maximum : write the quantity y in terms of one variable x (use the given condition), put the first derivative zero and check the second derivative.

F = K4(16q₁ − q₁²) ; R_eq = 20R₁ − R₁²20. In both cases the sum is fixed and the maximum comes when the two parts are equal.

26. Standard Result : a sin x + b cos x

y = a sin x + b cos x ⇒ dydx = a cos x − b sin x = 0 ⇒ tan x = ab

e.g. y = 3 sin x + 4 cos x : tan x = 3/4 ⇒ x = 37° ; d²y/dx² = −3 sin37° − 4 cos37° = −5 < 0 ⇒ y_max = 3(3/5) + 4(4/5) = 5 ; y_min = −5.

27. Scalars

• Scalar : physical quantity fully described by a single number (with unit) and having no direction, e.g. mass, temperature, volume, speed, charge.
• A scalar can be any real number — positive or negative, e.g. temperature 42 °C or −10 °C ; charge of proton +1.6 × 10⁻¹⁹ C, electron −1.6 × 10⁻¹⁹ C.
• Scalars add by simple algebraic sum : 10 C + 4 C = 14 C ; (+10 C) + (−6 C) = 4 C.
• Sign of a scalar shows its nature, not a direction : charges +10 C and −6 C are of opposite nature ; work W = +80 J ⇒ energy supplied, W = −60 J ⇒ energy withdrawn (like +10,000 and −8,000 in a bank account).
• Never drop the sign of a scalar, e.g. potential energy U = −Gm₁m₂r keeps its minus sign.

28. Vectors

• Vector : quantity having both magnitude and direction and which follows the laws of vector addition.
• Drawn as an arrow : length of the arrow corresponds to the magnitude, arrowhead shows the direction. Same direction but longer arrow ⇒ larger magnitude.

• In a vector, the sign shows direction : + ⇒ along +î (or +ĵ, +k̂), − ⇒ along −î (or −ĵ, −k̂). e.g. forces along x : F₁ = +10 N (towards +x), F₂ = −8 N (towards −x).
• Once the direction is drawn, the sign can be dropped and only the magnitude written : gravitational force F = −Gm₁m₂r² (minus ⇒ attraction) is drawn as two arrows towards each other, each of magnitude Gm₁m₂r².
• e.g. +ve direction to the right, velocity v to the right : a = −5 m/s² means 5 m/s² towards the left (opposite to v).
• Forces 8 N and 6 N : same direction ⇒ net 14 N ; opposite directions ⇒ net 2 N — the net depends on the directions.
• Electric current also has a direction, but currents 8 A and 6 A meeting at a junction always give 14 A, whatever their directions — current does not follow vector addition. Having a direction alone does not make a quantity a vector.

29. Unit Vector, Parallel & Antiparallel Vectors

• Unit vector  gives the direction of A→ ; its magnitude is 1 and it is unitless.
• e.g. A→ = 10 m towards east ⇒ A = 10 m, Â = (10 m towards east)/(10 m) = 1 towards east.

• Parallel vectors : same direction ⇒ Â = B̂ ; magnitudes A and B may or may not be equal.
• Antiparallel vectors : opposite directions ⇒ Â = −B̂.

30. Multiplying a Vector by a Scalar

• Any vector = magnitude × unit vector : A→ = AÂ.
• î, ĵ, k̂ → unit vectors along the x, y, z axes. F→ = (8î + 6ĵ + 3k̂) N means (8 N)î + (6 N)ĵ + (3 N)k̂.
• Positive scalar : direction stays the same, only the magnitude changes.

e.g. A→ = 4 m toward east ⇒ 3A→ = 12 m toward east ; A→ = 4î ⇒ 3A→ = 12î, A→/2 = 2î.

• Negative scalar : direction becomes opposite and the magnitude is multiplied (or divided) by the number.

A→ = 10 m toward east ⇒ −A→ = 10 m toward west. A→ = 10î ⇒ −A→ = 10(−î), −4A→ = 40(−î), −A→/2 = 5(−î).

31. Equal Vectors & Shifting a Vector

• A vector can be displaced parallel to itself : with the same magnitude and direction it is the same vector.
• This is used to bring vectors tail to tail (to measure the angle) or head to tail (to add them).

32. Angle between Two Vectors

• θ = angle between the vectors when their tails coincide (or their heads coincide) ; 0° ≤ θ ≤ 180° (take 60°, not 300°).
• Head of one on the tail of the other : the angle seen is not θ ; shift (extend) one vector ⇒ θ = 180° − (angle seen).

e.g. tails together at 30° ⇒ θ = 30° ; head of one on the tail of the other, 30° seen ⇒ θ = 180° − 30° = 150°.

33. Polygon Law of Vector Addition

• Vectors arranged in cyclic order (tail of the next on the head of the previous) : resultant goes from the tail of the first to the head of the last.

• Vectors in cyclic order forming a closed polygon ⇒ resultant zero : A→ + B→ + C→ + D→ + E→ = 0.
• Not in cyclic order? Reverse the wrong vector(s) by writing them with a minus sign. e.g. hexagon with B→ and E→ reversed : A→ + (−B→) + C→ + D→ + (−E→) + F→ = 0 ⇒ A→ + C→ + D→ + F→ = B→ + E→.
• Triangle : A→, B→, C→ head to tail all round ⇒ A→ + B→ + C→ = 0 ; C→ from the tail of A→ to the head of B→ ⇒ A→ + B→ = C→.

34. Equilibrium & Equal Coplanar Forces

• A particle is in equilibrium if the net force on it is zero :

• Forces in equilibrium, drawn head to tail, form a closed polygon.
• N equal, coplanar forces with zero resultant ⇒ the angles between adjacent forces are equal ; equal forces at equal angles cancel each other.

Head to tail, N equal forces form a regular polygon (3 forces ⇒ equilateral triangle) ; exterior angle = 360°/N.

35. Lami's Theorem

• If three coplanar forces acting at a point have zero resultant :

θ₁ = angle between F₂→ and F₃→ (opposite to F₁→) ; θ₂ between F₃→ and F₁→ ; θ₃ between F₁→ and F₂→.

Proof : head to tail the forces form a triangle with interior angles (180° − θ₁), (180° − θ₂), (180° − θ₃) ; sine rule + sin(180° − θ) = sinθ.

36. Triangle Law & Resultant of Two Vectors

• Triangle law : tail of B→ on the head of A→ ⇒ the resultant goes from the tail of A→ to the head of B→.

θ = angle between A→ and B→ : R² = (A + B cosθ)² + (B sinθ)² = A² + B²(cos²θ + sin²θ) + 2AB cosθ

37. Maximum & Minimum Resultant

• F₁ = 8 N, F₂ = 6 N : the resultant lies between 2 N and 14 N ⇒ 5 N, 13 N possible ; 1 N, 15 N not possible.

38. Resultant of Two Equal Vectors

A = B : R = √(2A²(1 + cosθ)) ; using 1 + cosθ = 2cos²(θ/2) ⇒ R = √(4A²cos²(θ/2))

• Unit vectors (|Â| = |B̂| = 1), using 1 − cosθ = 2sin²(θ/2) :

• cosθ positive ⇒ 0° ≤ θ < 90° ; cosθ negative ⇒ 90° < θ ≤ 180°.

39. Subtraction of Two Vectors

• Reverse B→ and add by the triangle law. Angle between A→ and B→ = θ ⇒ angle between A→ and −B→ = 180° − θ, and cos(180° − θ) = −cosθ.

• A→ = 8 m toward east, B→ = 6 m toward north : A→ + B→ = 8 m east + 6 m north ; A→ − B→ = 8 m east + 6 m south.
• 8 N, 6 N : |F₁→ − F₂→| = √(100 − 96 cosθ) = 2 N, √52 N, 10 N, √148 N, 14 N for θ = 0°, 60°, 90°, 120°, 180°.
• |F₁→ + F₂→| = 2|F₁→ − F₂→| : squaring ⇒ 10F₁F₂ cosθ = 3(F₁² + F₂²) ⇒ cosθ = 3(F₁² + F₂²)10F₁F₂ ; F₁ = F₂ ⇒ cosθ = 3/5.

40. Two Vectors of Equal Magnitude : |A→ + B→| = n|A→ − B→|

• |A→| = |B→| = F, angle θ between them. Square both sides of |A→ + B→| = n|A→ − B→| :

F² + F² + 2F² cosθ = n²(F² + F² − 2F² cosθ) ⇒ 1 + cosθ = n² − n² cosθ ⇒ (1 + n²) cosθ = n² − 1

• Unit vectors : |Â + B̂| = 2cos(θ/2), |Â − B̂| = 2sin(θ/2) ⇒ dividing :

41. Orthogonal Coordinate System & Position Vector

• Three mutually perpendicular axes X, Y, Z through the origin O (0, 0, 0) ; î, ĵ, k̂ → unit vectors along +X, +Y, +Z (−î, −ĵ, −k̂ along the negative axes).

Position vector of B w.r.t. A :

A ≡ (x₁, y₁, z₁), B ≡ (x₂, y₂, z₂) ; r→_A = x₁î + y₁ĵ + z₁k̂, r→_B = x₂î + y₂ĵ + z₂k̂

42. Rectangular Components in 2-D

• A vector A→ making angle θ with the x-axis is replaced by two perpendicular vectors along the axes (its components).

Right triangle : A_xA = cosθ, A_yA = sinθ

• e.g. A = 5 at θ = 37° : A_x = 5 × 45 = 4, A_y = 5 × 35 = 3 ⇒ A→ = 4î + 3ĵ.
• cos goes with the side of the angle : if θ is measured from the y-axis, then A_y = A cosθ and A_x = A sinθ (right figure).

e.g. vector at 30° with the y-axis, A_y = 2√3 : A cos30° = 2√3 ⇒ A = 4 ⇒ A_x = A sin30° = 2.

Signs in the four quadrants (θ = acute angle with the x-axis) :

43. Magnitude & Direction from Components

• Components A_x, A_y form a right triangle with A→ as hypotenuse :

θ = angle with the x-axis. e.g. v→ = 6î + 8ĵ : v = √(6² + 8²) = 10, tanθ = 8/6 = 4/3 ⇒ θ = 53°.

44. Resultant by the Component Method

• Resolve every vector along x and y ; add all x-components and all y-components (with signs).
• Then R = √(R_x² + R_y²), tanθ = R_y/R_x.

e.g. 12 N along −x, 16 N along +y and F below the +x-axis at θ have zero resultant : F cosθ = 12, F sinθ = 16 ⇒ F = √(12² + 16²) = 20 N, tanθ = 16/12 = 4/3.

45. Scalar (Dot) Product

A, B → magnitudes ; θ → angle between the vectors

• Result is only a number — a scalar, no direction.
• Sign depends only on cosθ : 0° ≤ θ < 90° ⇒ positive ; 90° < θ ≤ 180° ⇒ negative.
• A→ ⊥ B→ ⇒ A→ · B→ = 0.

A→ = A₁î + A₂ĵ + A₃k̂, B→ = B₁î + B₂ĵ + B₃k̂ :

e.g. (î + 2ĵ + 3k̂)·(4î + 5ĵ + 6k̂) = 4 + 10 + 18 = 32 ; (î − 2ĵ + 3k̂)·(4î − 5ĵ − 6k̂) = 4 + 10 − 18 = −4.

Uses :

• Perpendicular condition : P→ = î + 4ĵ + 2k̂ ⊥ Q→ = 6î − 4ĵ + mk̂ ⇒ 6 − 16 + 2m = 0 ⇒ m = 5.

46. Projection of a Vector (using Dot Product)

• Projection (component) of B→ on A→ : length B cosθ along Â.

(B cosθ)Â = (B cosθ)A→A = (AB cosθA²)A→

• The vector in the numerator and A² (or B²) in the denominator belong to the vector on which we project.
• e.g. 20 N at 37° with the x-axis : projection on the x-axis = (20 cos37°)î = 16î.

e.g. projection of Q→ = 3î − 4ĵ + 5k̂ on P→ = î + 2ĵ − 2k̂ : Q→·P→ = 3 − 8 − 10 = −15, P = 3 ⇒ −159(î + 2ĵ − 2k̂) = −53(î + 2ĵ − 2k̂).

47. Algebra of the Dot Product

• Dot product distributes and can be expanded like ordinary algebra : A→ · (3A→ + 2B→) = 3A→·A→ + 2A→·B→ = 3A² + 2AB cosθ.
• A→·B→ = B→·A→ ⇒ the two cross terms add up.

(A→ + B→)² = (A→ + B→)·(A→ + B→) = A→·A→ + A→·B→ + B→·A→ + B→·B→

• e.g. A = 3, B = 4, θ = 60° : A→·(3A→ + 2B→) = 27 + 24(½) = 39 ; (A→ + B→)² = 9 + 12 + 16 = 37.
• Method : if |A→ + B→| is given, square it to get AB cosθ ( = A→·B→ ), then expand the required product.

48. Position Vector Results

Point on the line joining two points :

S lies on R→ = Q→ − P→ with PS = b|R→| ⇒ S→ = P→ + bR→ = P→ + b(Q→ − P→)

Regular polygon with centre O :

• Vectors from the centre to all n vertices are equal and at equal angles ⇒ their sum is zero.

From vertex A : AX→ = AO→ + OX→ ; adding for the other (n − 1) vertices : (n − 1)AO→ + (0 − OA→) = (n − 1)AO→ + AO→

e.g. regular octagon, AO→ = 2î + 3ĵ − 4k̂ : AB→ + AC→ + … + AH→ = 8AO→ = 16î + 24ĵ − 32k̂.

49. Formula Summary – Basic Maths

50. Formula Summary – Vectors