Vector Analysis

🗺️ Cartesian Coordinate System

3D Cartesian Axes — Right-Hand Rule

Vector Representation

A vector A in Cartesian coordinates has components along x, y, z axes.

A = Ax·â_x + Ay·â_y + Az·â_z
|A| = √(Ax² + Ay² + Az²)

The unit vector â_A has magnitude = 1 and points in the direction of A.

â_A = A / |A|

Position & Distance Vectors

Position vector r_P is the directed distance from origin O to point P(x,y,z).

r_P = x·â_x + y·â_y + z·â_z

Distance vector from P to Q:

r_PQ = r_Q − r_P
= (xQ−xP)â_x + (yQ−yP)â_y + (zQ−zP)â_z

Vector Addition / Subtraction

Add (or subtract) component-by-component:

C = A + B
= (Ax+Bx)â_x + (Ay+By)â_y + (Az+Bz)â_z

Laws: Commutative (A+B = B+A), Associative, Distributive.

· Dot Product (Scalar Product) Lec 2

Geometric interpretation — projection of B onto A

Dot Product Definition

A · B = |A||B| cos θ_AB
= Ax·Bx + Ay·By + Az·Bz

Result is a scalar. θ is the smaller angle between A and B.

Unit Vector Rules

â_x·â_x = â_y·â_y = â_z·â_z = 1
â_x·â_y = â_y·â_z = â_z·â_x = 0

Parallel unit vectors dot to 1; perpendicular dot to 0.

Finding the Angle

cos θ = (A·B) / (|A|·|B|)
θ = arccos( A·B / |A||B| )

Used heavily in Tutorial problems to find interior angles of triangles.

📝 Tutorial Example — Interior Angles

A(1,3,−2), B(−2,4,5), C(0,−2,1). Find angle at A.

AB = (−3, 1, 7) → |AB| = √59
AC = (−1, −5, 3) → |AC| = √35
AB·AC = 3 − 5 + 21 = 19
cos θ_A = 19 / (√59·√35) → θ_A = 65.28°

× Cross Product (Vector Product) Lec 2

Cross Product — Right-Hand Rule & Determinant

Determinant Form

= (AyBz−AzBy)â_x − (AxBz−AzBx)â_y + (AxBy−AyBx)â_z

Unit Vector Cyclic Rules

â_x × â_y = â_z

â_y × â_z = â_x

â_z × â_x = â_y

â_y × â_x = −â_z

â_z × â_y = −â_x

â_x × â_z = −â_y

Mnemonic: x→y→z→x is positive; reverse is negative.

Area of Triangle

Area = ½ |r₂₃ × r₂₁|

The magnitude of the cross product = area of the parallelogram. Half of that is the triangle area.

💡 Tip: To find a unit vector ⊥ to a plane, compute the cross product of two vectors in the plane, then normalize.

∇ Del Operator & Vector Calculus Lec 4 · Tut 4

Del (∇) Operator

∇ = â_x ∂/∂x + â_y ∂/∂y + â_z ∂/∂z

Three applications: gradient (∇V), divergence (∇·A), curl (∇×A).

Gradient ∇V

Operates on a scalar → produces a vector pointing in direction of maximum increase.

∇V = (∂V/∂x)â_x + (∂V/∂y)â_y + (∂V/∂z)â_z

Key relation: E = −∇V (E-field is negative gradient of potential)

Divergence ∇·A

Operates on a vector → produces a scalar.

∇·A = ∂Ax/∂x + ∂Ay/∂y + ∂Az/∂z

Positive divergence = field spreading out (source). Negative = converging (sink). Zero = solenoidal.

Curl ∇×A

Operates on a vector → produces a vector. Measures rotation/circulation.

∇×A = | â_x â_y â_z |
| ∂/∂x ∂/∂y ∂/∂z|
| Ax Ay Az |

Zero curl = irrotational field.

Important Identities

∇×(∇V) = 0 (curl of gradient = 0)
∇·(∇×A) = 0 (div of curl = 0)

💡 Tip: These two identities are used to classify fields in Maxwell's equations.

Field Classifications

∇·A	∇×A	Type
= 0	= 0	Irrotational + Solenoidal
≠ 0	= 0	Irrotational only
= 0	≠ 0	Solenoidal only
≠ 0	≠ 0	Neither

📝 Tutorial Example — Gradient

Find ∇V for V = e^(2x+3y) · cos(5z) at P(0.1, −0.2, 0.4)

∂V/∂x = 2e^(2x+3y)cos(5z) → at P: −0.5579
∂V/∂y = 3e^(2x+3y)cos(5z) → at P: −0.8369
∂V/∂z = −5e^(2x+3y)sin(5z) → at P: −3.0476
∇V|_P = −0.558â_x − 0.837â_y − 3.048â_z