Vectors →

Magnitude aur direction — dono milke ek vector banate hain. Physics se computer graphics tak — sab jagah vectors!

✈️ Pilot ki Kahani

Ek pilot apna plane North mein 200 km/h se fly kar raha hai. Tabhi ek crosswind 50 km/h East ki taraf aati hai. Plane actually kahan jayega? Seedha North nahi — thoda Northeast ki taraf! Kyunki wind aur plane ki apni speed dono mil jaate hain — aur resultant direction calculate karna hota hai. Yeh exactly vector addition hai. Physics mein forces, velocity, acceleration — sab vectors hain. Math mein inhe formally study karte hain.

1. Scalar vs Vector

Scalar

Sirf magnitude (size): Temperature, mass, speed, distance. Koi direction nahi.

Examples: 5 kg, 30°C, 100 km

Vector

Magnitude + Direction: Velocity, Force, Displacement. Arrow se represent karte hain.

Examples: 60 km/h North, 5N downward

Vector Components in 3D x z y ⃗a aₓ (x-component) a_z ⃗a = aₓ î + aᵧ ĵ + a_z k̂ |⃗a| = √(aₓ²+aᵧ²+a_z²)

2. Position Vector aur Direction Cosines

Position vector of P(x,y,z): OP⃗ = xî + yĵ + zk̂ Direction cosines (l, m, n): l = cos α = x/r, m = cos β = y/r, n = cos γ = z/r where r = |OP⃗| = √(x²+y²+z²) Important: l² + m² + n² = 1 (always!)

3. Scalar (Dot) Product

⃗a·⃗b = |⃗a||⃗b|cos θ = a₁b₁ + a₂b₂ + a₃b₃

Key result: ⃗a ⊥ ⃗b ⟺ ⃗a·⃗b = 0

î·î = ĵ·ĵ = k̂·k̂ = 1; î·ĵ = ĵ·k̂ = k̂·î = 0

cos θ = (⃗a·⃗b)/(|⃗a||⃗b|) Projection of ⃗a on ⃗b = (⃗a·⃗b)/|⃗b| Work done = F⃗·d⃗ (physics application)

4. Vector (Cross) Product

⃗a × ⃗b = |⃗a||⃗b|sin θ n̂ (n̂ = unit normal using right-hand rule)

Key result: ⃗a ∥ ⃗b ⟺ ⃗a × ⃗b = ⃗0

î×ĵ=k̂, ĵ×k̂=î, k̂×î=ĵ

⃗a × ⃗b = | î ĵ k̂ | | a₁ a₂ a₃| | b₁ b₂ b₃| = î(a₂b₃-a₃b₂) - ĵ(a₁b₃-a₃b₁) + k̂(a₁b₂-a₂b₁) |⃗a × ⃗b| = area of parallelogram formed by ⃗a and ⃗b Area of triangle = ½|⃗a × ⃗b|

✅ ⃗a = 2î+3ĵ-k̂, ⃗b = î-ĵ+2k̂. Find ⃗a×⃗b.

= î(3×2-(-1)×(-1)) - ĵ(2×2-(-1)×1) + k̂(2×(-1)-3×1)

= î(6-1) - ĵ(4+1) + k̂(-2-3)

= 5î - 5ĵ - 5k̂

5. Practice Questions

Q1. Find angle between ⃗a=2î-ĵ+2k̂ and ⃗b=3î+4ĵ-5k̂.

Solution

⃗a·⃗b=6-4-10=-8. |⃗a|=√(4+1+4)=3. |⃗b|=√(9+16+25)=√50. cosθ=-8/(3√50). θ=cos⁻¹(-8/3√50)

Q2. Show vectors 2î-3ĵ+4k̂ and 4î-6ĵ+8k̂ are parallel.

Solution

Second = 2 × first. Parallel vectors are scalar multiples. Cross product = 0. OR direction ratios proportional: 2/4=3/6=4/8=1/2. ✓

Q3. Find unit vector in direction of ⃗a = î+2ĵ-3k̂.

Solution

|⃗a|=√(1+4+9)=√14. â = (î+2ĵ-3k̂)/√14