Vectors | IISER Smart Prep

1. Core Concept

A vector represents a quantity with both magnitude and direction, obeying the triangle law of addition. Vectors act as the bridge between pure algebra and 3D geometry, allowing us to describe lengths, angles, areas, and spatial orientations purely through equations.

Types of Vectors

Zero Vector: Magnitude = 0, direction is indeterminate.
Unit Vector: Magnitude = 1. Used to specify direction.
Equal Vectors: Have the same magnitude and the same direction.
Negative Vectors: Have the same magnitude but exactly opposite direction (e.g., AB = -BA).
Parallel/Collinear Vectors: Vectors whose directions are the same or opposite. Mathematically: a⃗ = λb⃗.

2. Key Formulas

â = a⃗ / |a⃗|

Unit Vector: Magnitude of 1, defines direction.

Component Form

A vector is written in terms of unit vectors along axes: a⃗ = xi + yj + zk

Magnitude: |a⃗| = √(x² + y² + z²)

Position Vector

The position vector of a point P(x, y, z) with respect to the origin O(0,0,0) is:

OP⃗ = xi + yj + zk

Direction Cosines (l, m, n):

Cosines of angles (α, β, γ) with axes: l = cosα, m = cosβ, n = cosγ.
Crucial Relation: l² + m² + n² = 1.

Internal: r⃗ = (mb⃗ + na⃗) / (m+n)
External: r⃗ = (mb⃗ - na⃗) / (m-n)

Section Formula for ratio m:n.

a⃗ · b⃗ = |a⃗||b⃗| cosθ = a₁b₁ + a₂b₂ + a₃b₃

Dot (Scalar) Product.

a⃗ × b⃗ = |a⃗||b⃗| sinθ n̂

Cross (Vector) Product. (Result is perpendicular to both).

Vector Projection of a on b = (a⃗ · b̂) b̂

Vector form of the projection.

3. Important Results

Orthogonal: a⃗ ⊥ b⃗ ⇔ a⃗ · b⃗ = 0.
Collinear (Parallel): a⃗ ∥ b⃗ ⇔ a⃗ × b⃗ = 0 OR a⃗ = λb⃗.
Lagrange’s Identity: |a⃗ × b⃗|² + (a⃗ · b⃗)² = |a⃗|²|b⃗|².

Areas:

Triangle: ½ |a⃗ × b⃗| (adjacent sides).
Parallelogram: |a⃗ × b⃗| (adjacent) OR ½ |d₁⃗ × d₂⃗| (diagonals).

4. Conceptual Insights

Visual Intuition:

Dot Product: Tells you "how much" of one vector acts in the direction of another (like a shadow/projection).

Cross Product: Provides the "Normal" vector perpendicular to the plane formed by both vectors, and its magnitude is the area spanned by them.

[a⃗ b⃗ c⃗] = a⃗ · (b⃗ × c⃗)

Scalar Triple Product: Represents the Volume of a Parallelepiped formed by vectors a, b, and c.

Squaring a Vector:

Always remember that |a⃗ ± b⃗|² = |a⃗|² + |b⃗|² ± 2(a⃗ · b⃗). This is the primary way to evaluate sums and differences of vector magnitudes.

5. Common Mistakes

Vertices vs. Sides: If given position vectors of vertices A, B, C, the area is not ½|a⃗ × b⃗|. You must find side vectors first (e.g., AB⃗ = b⃗ - a⃗).
Cross Product Order: a⃗ × b⃗ = -(b⃗ × a⃗). The order strictly changes the sign.
Cancellation Law Failure: a⃗ · b⃗ = a⃗ · c⃗ does not imply b⃗ = c⃗. It only means a⃗ is perpendicular to the vector (b⃗ - c⃗).

6. IAT Exam Focus Points

Perpendicular Vectors:

Given a⃗ and b⃗, finding a vector of magnitude 5 perpendicular to both: Use the cross product, normalize it to a unit vector, then multiply by 5: ±5 * (a⃗ × b⃗) / |a⃗ × b⃗|.

Solving Vector Equations:

If given x⃗ × a⃗ = b⃗, take the dot or cross product of the entire equation with a⃗ to isolate x⃗.

Angle Bisectors:

The vector bisecting the angle between a⃗ and b⃗ is λ(â + b̂). For the external bisector, use λ(â - b̂).

Direction Cosines Trap:

If a line makes angles α, β, γ, remember sin²α + sin²β + sin²γ = 2 (This is a quick way to bypass the standard l²+m²+n²=1 identity).

7. Practice Mock Test

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