Class 11 · Vectors JEE Main Quick Notes
VECTOR ALGEBRA
Class 11 · Handwritten Notes
JEE Main Focused − Concepts + Formulas
What's inside?
• Scalars & Vectors, types • Dot product & projections
• Vector addition laws • Cross product & areas
• Section formula • Scalar & vector triple product
• Direction cosines/ratios • JEE formula sheet + tricks
1. Scalar vs Vector
• Scalar: has only magnitude. E.g. mass, speed, temperature, work.
• Vector: has magnitude and direction, obeys triangle law of addition. E.g. displacement,
velocity, force.
−−→
A vector AB is written as ⃗a or a, with magnitude |⃗a| or a.
2. Types of Vectors
−→
• Zero (Null) vector: magnitude = 0, direction not defined. AA = ⃗0.
⃗a
• Unit vector: magnitude = 1. â =
|⃗a|
• Equal vectors: same magnitude & same direction.
• Collinear (parallel) vectors: parallel to the same line, irrespective of magnitude/direction.
• Coplanar vectors: lie in (or parallel to) the same plane.
• Coinitial vectors: same starting point.
• Negative of a vector: same magnitude, opposite direction, −⃗a.
−−→
• Position vector: vector from origin O to point P , OP .
1
, Class 11 · Vectors JEE Main Quick Notes
3. Addition of Vectors
Triangle Law
If two vectors are represented (in magnitude & direction) by two sides of a triangle taken
in order, the resultant is the third side taken in reverse order:
−−→ −−→ −→
AB + BC = AC
Parallelogram Law
If two vectors are represented by adjacent sides of a parallelogram from a common
point, their sum is the diagonal from that point.
p
|⃗a + ⃗b| = a2 + b2 + 2ab cos θ
Properties of Vector Addition
Key Properties
• Commutative: ⃗a + ⃗b = ⃗b + ⃗a
• Associative: (⃗a + ⃗b) + ⃗c = ⃗a + (⃗b + ⃗c)
• Identity: ⃗a + ⃗0 = ⃗a
• Inverse: ⃗a + (−⃗a) = ⃗0
⋆ JEE Trick ⋆
For |⃗a + ⃗b| max value = a + b (same direction, θ = 0◦ ); min value = |a − b| (opposite,
θ = 180◦ ).
4. Multiplication by a Scalar
If m is a scalar, m⃗a is a vector with magnitude |m||⃗a|, direction same as ⃗a if m > 0,
opposite if m < 0.
5. Section Formula
−→
Position vector of point dividing AB
−−→
Point R divides AB in ratio m : n.
m⃗b + n⃗a
Internally: ⃗r =
m+n
m⃗b − n⃗a
Externally: ⃗r =
m−n
⃗a + ⃗b
Midpoint (m = n = 1): ⃗r =
2
2
VECTOR ALGEBRA
Class 11 · Handwritten Notes
JEE Main Focused − Concepts + Formulas
What's inside?
• Scalars & Vectors, types • Dot product & projections
• Vector addition laws • Cross product & areas
• Section formula • Scalar & vector triple product
• Direction cosines/ratios • JEE formula sheet + tricks
1. Scalar vs Vector
• Scalar: has only magnitude. E.g. mass, speed, temperature, work.
• Vector: has magnitude and direction, obeys triangle law of addition. E.g. displacement,
velocity, force.
−−→
A vector AB is written as ⃗a or a, with magnitude |⃗a| or a.
2. Types of Vectors
−→
• Zero (Null) vector: magnitude = 0, direction not defined. AA = ⃗0.
⃗a
• Unit vector: magnitude = 1. â =
|⃗a|
• Equal vectors: same magnitude & same direction.
• Collinear (parallel) vectors: parallel to the same line, irrespective of magnitude/direction.
• Coplanar vectors: lie in (or parallel to) the same plane.
• Coinitial vectors: same starting point.
• Negative of a vector: same magnitude, opposite direction, −⃗a.
−−→
• Position vector: vector from origin O to point P , OP .
1
, Class 11 · Vectors JEE Main Quick Notes
3. Addition of Vectors
Triangle Law
If two vectors are represented (in magnitude & direction) by two sides of a triangle taken
in order, the resultant is the third side taken in reverse order:
−−→ −−→ −→
AB + BC = AC
Parallelogram Law
If two vectors are represented by adjacent sides of a parallelogram from a common
point, their sum is the diagonal from that point.
p
|⃗a + ⃗b| = a2 + b2 + 2ab cos θ
Properties of Vector Addition
Key Properties
• Commutative: ⃗a + ⃗b = ⃗b + ⃗a
• Associative: (⃗a + ⃗b) + ⃗c = ⃗a + (⃗b + ⃗c)
• Identity: ⃗a + ⃗0 = ⃗a
• Inverse: ⃗a + (−⃗a) = ⃗0
⋆ JEE Trick ⋆
For |⃗a + ⃗b| max value = a + b (same direction, θ = 0◦ ); min value = |a − b| (opposite,
θ = 180◦ ).
4. Multiplication by a Scalar
If m is a scalar, m⃗a is a vector with magnitude |m||⃗a|, direction same as ⃗a if m > 0,
opposite if m < 0.
5. Section Formula
−→
Position vector of point dividing AB
−−→
Point R divides AB in ratio m : n.
m⃗b + n⃗a
Internally: ⃗r =
m+n
m⃗b − n⃗a
Externally: ⃗r =
m−n
⃗a + ⃗b
Midpoint (m = n = 1): ⃗r =
2
2
