Chapter 32 · Class 11-12 & JEE Main/Adv

Vectors

67 topics in 15 parts

Full lessonClass 12 Mathematics · Vectors and 3D Geometry Practice4 practice questions

Dot product Cross product magnitude Magnitude of a Vector Unit Vector Position Vector Vector Joining Two Points

Key ideas

Conceptual Foundation & Core Theory

In 3D Euclidean space, a Plane is a flat two-dimensional surface that extends infinitely in all directions. A plane is uniquely positioned in space if we know either (1) its perpendicular distance from the origin and its normal direction, or (2) a point lying on the plane and a normal vector perpendicular to it.

General Cartesian Equation of a Plane: Every first-degree equation in x, y, z represents a plane: Ax + By + Cz + D = 0 Crucially, the coefficients <A, B, C> represent the Direction Ratios of the Normal Vector perpendicular to the plane!

Intercept Form: If a plane cuts off intercepts a, b, c from the coordinate axes X, Y, Z respectively, its equation is: x / a + y / b + z / c = 1

Standard Formulas & Governing Conditions

// Plane Formulations in 3D Space

1. General Cartesian Form:
   Ax + By + Cz + D = 0  (Normal vector n = Aî + Bĵ + Ck̂)

2. Vector Form Passing through a with Normal n:
   (r - a) · n = 0  ⟺  r · n = d

3. Normal Form (Unit Normal n̂ and Perpendicular Distance p from Origin):
   r · n̂ = p  ⟺  lx + my + nz = p

4. Intercept Form (Intercepts a, b, c on axes):
   x/a + y/b + z/c = 1

5. Perpendicular Distance of Point (x1, y1, z1) from Plane Ax + By + Cz + D = 0:
   d = | Ax1 + By1 + Cz1 + D | / √(A² + B² + C²)

6. Distance Between Two Parallel Planes Ax + By + Cz + D1 = 0 and Ax + By + Cz + D2 = 0:
   d = | D1 - D2 | / √(A² + B² + C²)

Vectors

Standard Forms of Plane Equations

Plane Equation Form	Formula	Geometrical Information Conveyed
General Form	Ax + By + Cz + D = 0	Normal direction ratios are <A, B, C>
Normal Form	lx + my + nz = p	p is exact perpendicular distance from origin; <l, m, n> are DCs
Intercept Form	x/a + y/b + z/c = 1	Plane cuts X at (a,0,0), Y at (0,b,0), Z at (0,0,c)
Three-Point Form	\| x-x1 y-y1 z-z1 ; x2-x1 y2-y1 z2-z1 ; x3-x1 y3-y1 z3-z1 \| = 0	Plane passing through three non-collinear points

Step-by-Step Solved Problem (JEE Main Standard)

Problem: Find the perpendicular distance between the parallel planes 2x - 2y + z + 3 = 0 and 4x - 4y + 2z + 5 = 0.

Step 1: Match Coefficients:
Multiply Plane 1 by 2: 4x - 4y + 2z + 6 = 0.
Plane 2: 4x - 4y + 2z + 5 = 0.

Step 2: Apply Parallel Planes Distance Formula:
Here A = 4, B = -4, C = 2, D₁ = 6, D₂ = 5.
d = |D₁ - D₂| / √(A² + B² + C²)
d = |6 - 5| / √(4² + (-4)² + 2²) = 1 / √(16 + 16 + 4) = 1 / √36 = 1/6 units.

Vectors

Analytical Comparison & Distinction Matrix

Analytical Method	Standard Representation	Parametric / Invariant Framework
Algebraic Nature	Explicit equations relating spatial coordinates	Parameterized using scalar parameters (λ, μ) or vectors
Symmetry Usage	Relies on coordinate expansion	Exploits vector dot/cross products for rapid simplification
Computational Efficiency	Direct for intercept evaluation	Superior for angles, orthogonal projections, and distances
Common Mistake	Sign confusion in determinants	Omitting boundary condition checks

Vectors

Property	Direction Cosines (l, m, n)	Direction Ratios (a, b, c)
Definition	Actual cosines of angles made with axes (cos α, cos β, cos γ)	Any set of numbers proportional to direction cosines
Normalization	Strictly normalized: l² + m² + n² = 1	Not normalized; can be multiplied by any constant k ≠ 0
Uniqueness	Unique for a given directed line (up to ± sign)	Infinitely many sets for the same line
Unit Vector	lî + mĵ + nk̂ is always a UNIT VECTOR of magnitude 1	aî + bĵ + ck̂ has magnitude √(a² + b² + c²)

Step-by-Step Solved Problem (JEE Main Standard)

Problem: A line makes angles of 60° and 45° with the positive X and Y axes respectively. Find the angle it makes with the positive Z axis.

Step 1: Identify Direction Angles:
α = 60° ⟹ l = cos 60° = 1/2.
β = 45° ⟹ m = cos 45° = 1/√2.
Let angle with Z axis be γ ⟹ n = cos γ.

Step 2: Apply Fundamental Identity:
l² + m² + n² = 1 ⟹ (1/2)² + (1/√2)² + cos²γ = 1
1/4 + 1/2 + cos²γ = 1 ⟹ 3/4 + cos²γ = 1
cos²γ = 1 - 3/4 = 1/4 ⟹ cos γ = ± 1/2.

Step 3: Deduce Angles:
If cos γ = 1/2 ⟹ γ = 60° (or π/3).
If cos γ = -1/2 ⟹ γ = 120° (or 2π/3).

Conceptual Foundation & Core Theory

In three-dimensional space, two lines can either be parallel, intersecting, or skew (non-parallel lines that never intersect because they lie in different planes). For two straight lines to be coplanar, they must lie simultaneously on a single flat plane. In that plane, they must either be parallel or intersect at a single unique point.

The Vector Condition: Consider two lines defined in vector form: Line 1: r = a₁ + λb₁ Line 2: r = a₂ + μb₂ Here, a₁ and a₂ are position vectors of known points on the lines, while b₁ and b₂ are vectors parallel to the lines. The vector connecting point A on Line 1 to point B on Line 2 is (a₂ - a₁). For both lines to reside on the same plane, (a₂ - a₁) must be completely coplanar with b₁ and b₂. Because the cross-product vector (b₁ × b₂) is normal (perpendicular) to the plane containing both lines, the scalar triple product must strictly vanish: (a₂ - a₁) · (b₁ × b₂) = 0

The Cartesian Determinant Test: Expressing the lines in symmetric Cartesian form: (x - x₁)/a₁ = (y - y₁)/b₁ = (z - z₁)/c₁ and (x - x₂)/a₂ = (y - y₂)/b₂ = (z - z₂)/c₂ The necessary and sufficient condition for coplanarity is that the 3×3 determinant equals zero: | (x₂ - x₁) (y₂ - y₁) (z₂ - z₁) | | a₁ b₁ c₁ | = 0 | a₂ b₂ c₂ |

Standard Formulas & Governing Conditions

// 3D Geometry: Coplanarity & Intersection Governing Equations

1. Vector Condition for Coplanarity:
   (a2 - a1) · (b1 × b2) = 0   [Scalar Triple Product = 0]

2. Cartesian Condition for Coplanarity (Determinant Form):
   | x2 - x1   y2 - y1   z2 - z1 |
   |   a1        b1        c1    | = 0
   |   a2        b2        c2    |

3. Equation of the Plane Containing Both Coplanar Lines:
   | x - x1   y - y1   z - z1 |
   |   a1       b1       c1   | = 0
   |   a2       b2       c2   |

4. Shortest Distance Between Non-Coplanar (Skew) Lines:
   d = | (a2 - a1) · (b1 × b2) | / | b1 × b2 |

Vectors

Relative Spatial Orientations of Two Lines in 3D Space

Orientation Type	Coplanarity Status	Shortest Distance (d)	Direction Ratios Condition
Intersecting Lines	Always Coplanar (Share 1 plane)	d = 0 (Unique point of contact)	Non-proportional direction ratios (b1 ≠ k b2)
Parallel Lines	Always Coplanar (Share 1 plane)	d = \|(a2 - a1) × b\| / \|b\|	Direction ratios are strictly proportional: a1/a2 = b1/b2 = c1/c2
Skew Lines	Non-Coplanar (Cannot share a plane)	d = \|(a2-a1)·(b1×b2)\| / \|b1×b2\| > 0	Non-parallel and non-intersecting lines in space
Coincident Lines	Coplanar (Identical lines)	d = 0 (Infinite points of contact)	Same direction ratios and share all points

Step-by-Step Solved Problem (JEE Advanced Standard)

Problem: Show that the lines L₁: (x - 1)/2 = (y - 2)/3 = (z - 3)/4 and L₂: (x - 2)/3 = (y - 3)/4 = (z - 4)/5 are coplanar, and find their point of intersection.

Step 1: Identify Coordinates and Direction Ratios:
(x₁, y₁, z₁) = (1, 2, 3) and (a₁, b₁, c₁) = (2, 3, 4)
(x₂, y₂, z₂) = (2, 3, 4) and (a₂, b₂, c₂) = (3, 4, 5)

Step 2: Evaluate the Determinant:
| 1 1 1 ; 2 3 4 ; 3 4 5 | = 1(-1) - 1(-2) + 1(-1) = 0.
Determinant = 0 ⟹ Lines are COPLANAR!

Step 3: Point of Intersection:
Equating coordinates gives P = (-1, -1, -1).

Conceptual Foundation & Core Theory

By Euclid's fundamental postulate extended into three dimensions, two distinct points in space uniquely determine a straight line. If point A(x₁, y₁, z₁) and point B(x₂, y₂, z₂) lie on the line, the vector joining them AB = (x₂ - x₁)î + (y₂ - y₁)ĵ + (z₂ - z₁)k̂ provides the line's direction vector.

Cartesian Equation (Two-Point Form): (x - x₁) / (x₂ - x₁) = (y - y₁) / (y₂ - y₁) = (z - z₁) / (z₂ - z₁) = λ Notice that the denominators (x₂ - x₁), (y₂ - y₁), (z₂ - z₁) directly represent the direction ratios of the line.

Standard Formulas & Governing Conditions

// Line Through Two Points Equations

1. Cartesian Two-Point Form:
   (x - x1)/(x2 - x1) = (y - y1)/(y2 - y1) = (z - z1)/(z2 - z1) = λ

2. Vector Two-Point Form:
   r = a + λ(b - a)  =  (1 - λ)a + λb

3. Direction Cosines of Line AB:
   l = (x2 - x1) / |AB|,  m = (y2 - y1) / |AB|,  n = (z2 - z1) / |AB|
   where |AB| = √[ (x2-x1)² + (y2-y1)² + (z2-z1)² ]

Vectors

One-Point Form vs Two-Point Form

Characteristic	Line Through Point & Direction Vector	Line Through Two Points
Required Data	One point (x1, y1, z1) and direction ratios <a, b, c>	Two coordinates (x1, y1, z1) and (x2, y2, z2)
Direction Ratios	Given directly as <a, b, c>	Calculated as <x2 - x1, y2 - y1, z2 - z1>
Vector Equation	r = a + λb	r = a + λ(b - a)
Midpoint Coordinate	Requires specifying λ = distance/(2\|b\|)	Direct midpoint formula: ((x1+x2)/2, (y1+y2)/2, (z1+z2)/2)

Step-by-Step Solved Problem (Board Exam Standard)

Problem: Find the Cartesian and vector equations of the line passing through points A(3, 4, -7) and B(1, -1, 6). Determine whether the point C(5, 9, -20) lies on this line.

Step 1: Direction Ratios:
<a, b, c> = <x₂ - x₁, y₂ - y₁, z₂ - z₁> = <1 - 3, -1 - 4, 6 - (-7)> = <-2, -5, 13>.

Step 2: Equations of Line AB:
• Vector Form: r = (3î + 4ĵ - 7k̂) + λ(-2î - 5ĵ + 13k̂)
• Cartesian Form: (x - 3) / -2 = (y - 4) / -5 = (z + 7) / 13 = λ

Step 3: Test Point C(5, 9, -20):
Substitute (5, 9, -20) into the Cartesian equation:
(5 - 3) / -2 = 2 / -2 = -1
(9 - 4) / -5 = 5 / -5 = -1
(-20 + 7) / 13 = -13 / 13 = -1
Since λ = -1 for all three coordinates, point C lies on the line (collinear)!

Checkpoint

1. What are the direction ratios of the normal to the plane 3x - 4y + 5z = 12?

2. What is the value of sin²α + sin²β + sin²γ where α, β, γ are the direction angles of a line?

3. If two non-parallel lines in space have a shortest distance of zero, they are:

4. What are the direction ratios of the line passing through (2, 3, 5) and (4, -1, 8)?

All 67 topics in Vectors

Coordinate Axes & Coordinate Planes in Three-Dimensional Space
- Coordinate Axes & Coordinate Planes
- Coordinates of a Point in Space
- Distance Between Two Points
Evolution of Vector Concept
- Definition
- Direction Cosines
- Position Vector
Types of Vectors
- Co-initial Vectors
- Collinear Vectors
- Coplanar Vectors
- Equal Vectors
- Free Vectors
- Like & Unlike Vectors
- Localised Vectors
- Negative of a Vector
- Non-collinear Vectors
- Parallel Vectors
- Unit Vector
- Zero Vector
Vector Operations & Components
- Basic Vector Algebra
- Vector Addition
- Properties of Vector Addition
- Components of a Vector
- Scalar Multiplication
Linear Combination, Linear Independence & Linear Dependence
- Linear Combination
- Linearly Dependent Vectors
- Linearly Independent Vectors
- Coplanarity of Vectors
- Condition of Coplanarity of Four Points
Dot (Scalar) Product
- Geometrical Interpretation of Scalar Product
- Physical Interpretation of Scalar Product
- Properties of Dot (Scalar) Product
Applications of Dot (Scalar) Product
- Cosine Rule Using Dot Product
- Finding Angle Between Two Vectors
- Projection of a Vector
- Resolving a Vector Along & Perpendicular to Another Vector
Vector (or Cross) Product of Two Vectors
- Geometrical Interpretation of Cross Product
- Physical Interpretation as a Moment of Force
- Properties of Cross Product
- Area Using Cross Product
Scalar Triple Product
- Geometric Interpretation
- Properties of Scalar Triple Product
- Volume of Tetrahedron
Vector Triple Product & Higher Products
- Lagrange's Identity
- Scalar Product of Four Vectors
- Vector Product of Four Vectors
Reciprocal System of Vectors
- Reciprocal System of Vectors
Vector Geometry
- Vector Joining Two Points
- Section Formula
- Internal Division
- External Division
- Vector Along the Bisector of Two Vectors
- Parallelograms & Polygons
- Triangle Geometry
- Vector Optimization
Straight Lines in Three-Dimensional Space
- Equation Through a Point Parallel to a Vector
- Equation Through Two Points
- Angle Between Two Lines
- Perpendicular Distance of a Point From a Line
- Foot of Perpendicular & Reflection in a Line
- Shortest Distance Between Two Lines
- Concurrency of Lines
Planes in Three-Dimensional Space
- Equation of a Plane
- Angle Between Two Planes
- Line of Intersection of Two Planes
Relations Between Lines & Planes
- Angle Between a Line & a Plane
- Projection of a Line on a Plane
- Reflection of a Line in a Plane