Vectors and Scalars Class 11: All Formulas, Types & Formula Sheet with Free PDF Download (Physics, JEE & NEET)
Vectors & Scalars Class 11 Formula Sheet — Competishun
Vectors and Scalars Class 11: All Formulas, Types & Formula Sheet with Free PDF Download (Physics, JEE & NEET)
Vectors and Scalars is one of the first topics in Class 11 physics, and it is the language the rest of physics is written in. Almost everything, from motion and force to electric and magnetic fields, is described using vectors. Get comfortable with them early, and the whole subject becomes easier.
This page gives you the complete Vectors and Scalars formula sheet for Class 11 in one place: scalar vs vector, types of vectors, addition and the parallelogram law, resolution into components, unit vectors, dot product and cross product, with worked examples. Download the free PDF below and keep it handy for revision.
Download the Vectors & Scalars Formula Sheet PDF
Get all Vectors and Scalars formulas, types, addition, resolution, dot and cross product in one clean PDF, free. Perfect for Class 11, JEE and NEET revision.
Download Free PDFScalar vs Vector: The Basic Difference
A scalar has only magnitude. A vector has both magnitude and direction.
| Property | Scalar | Vector |
|---|---|---|
| Has | Magnitude only | Magnitude + direction |
| Added by | Simple arithmetic | Vector laws |
| Examples | Mass, speed, time, energy, temperature | Displacement, velocity, force, acceleration |
| A quick test: if direction matters, it is a vector. Speed is scalar, velocity is a vector. | ||
Types of Vectors
| Type | Meaning |
|---|---|
| Unit Vector | Vector of magnitude 1, shows direction only (Â = A/|A|) |
| Zero / Null Vector | Vector with zero magnitude, no specific direction |
| Equal Vectors | Same magnitude and same direction |
| Negative Vector | Same magnitude, opposite direction |
| Collinear Vectors | Along the same or parallel lines |
| Coplanar Vectors | Lie in the same plane |
| Position Vector | Vector from origin to a point |
| Displacement Vector | Vector from initial to final position |
| Unit vectors î, ĵ, k̂ point along the x, y and z axes and are the backbone of component form. | |
Vector Addition Formulas
| Law / Case | Formula |
|---|---|
| Triangle law | Place head to tail, resultant closes the triangle |
| Parallelogram law (magnitude) | R = √(A² + B² + 2AB cosθ) |
| Direction of resultant | tanα = (B sinθ) / (A + B cosθ) |
| Same direction (θ=0) | R = A + B (maximum) |
| Opposite direction (θ=180°) | R = |A − B| (minimum) |
| Perpendicular (θ=90°) | R = √(A² + B²) |
| The resultant of two vectors always lies between (A + B) and |A − B|. | |
Resolution of Vectors (Components)
Any vector can be broken into perpendicular components, which makes calculations far easier.
| Quantity | Formula |
|---|---|
| x-component | Aₓ = A cosθ |
| y-component | Aᵧ = A sinθ |
| Magnitude from components | A = √(Aₓ² + Aᵧ²) |
| Direction from components | tanθ = Aᵧ / Aₓ |
| Component form | A⃗ = Aₓ î + Aᵧ ĵ + A_z k̂ |
| Magnitude in 3D | |A⃗| = √(Aₓ² + Aᵧ² + A_z²) |
| Resolving into components is the standard method to add several vectors accurately. | |
Dot Product (Scalar Product)
| Concept | Formula |
|---|---|
| Definition | A⃗ · B⃗ = AB cosθ (a scalar) |
| Component form | A⃗ · B⃗ = AₓBₓ + AᵧBᵧ + A_zB_z |
| Angle between vectors | cosθ = (A⃗ · B⃗) / (AB) |
| Unit vector rules | î·î = ĵ·ĵ = k̂·k̂ = 1, î·ĵ = 0 |
| Physics use | Work = F⃗ · d⃗ |
| The dot product is maximum when vectors are parallel (θ=0) and zero when perpendicular (θ=90°). | |
Cross Product (Vector Product)
| Concept | Formula |
|---|---|
| Definition | |A⃗ × B⃗| = AB sinθ (a vector) |
| Direction | Perpendicular to both, by right-hand rule |
| Anti-commutative | A⃗ × B⃗ = − (B⃗ × A⃗) |
| Parallel vectors | A⃗ × A⃗ = 0 |
| Unit vector rules | î×ĵ = k̂, ĵ×k̂ = î, k̂×î = ĵ |
| Physics use | Torque τ⃗ = r⃗ × F⃗, area = |A⃗ × B⃗| |
| The cross product is maximum when vectors are perpendicular (θ=90°) and zero when parallel. | |
Worked Examples
| Problem | Solution |
|---|---|
| Resultant of 3 and 4 at 90° | R = √(3² + 4²) = 5 |
| Components of 10 at 30° | Aₓ = 10cos30 = 8.66, Aᵧ = 10sin30 = 5 |
| Dot: (2î+3ĵ)·(î+4ĵ) | 2×1 + 3×4 = 14 |
| Magnitude of 3î+4ĵ | √(3² + 4²) = 5 |
| Cross magnitude, A=2,B=3,θ=90° | |A×B| = 2×3×sin90 = 6 |
| Notice the 3-4-5 pattern appears often, it is worth memorising. | |
Common Mistakes to Avoid
- Adding vectors like scalars: You cannot just add magnitudes, direction matters
- Dot vs cross: Dot gives a scalar (cosθ), cross gives a vector (sinθ)
- Wrong component: The component along the axis uses cos, perpendicular uses sin
- Forgetting direction: A resultant needs both magnitude and direction (the angle α)
Why Vectors Matter for JEE & NEET
- Used everywhere: Kinematics, forces, fields and rotation all rely on vectors
- Direct questions: Dot and cross product questions appear in JEE and NEET regularly
- Foundation topic: Weak vectors make all of mechanics harder
- Quick marks: Once formulas are clear, vector problems are fast and reliable
How to Use This Formula Sheet
- Master the resultant formula: R = √(A²+B²+2AB cosθ) is used constantly
- Practise resolution: Breaking vectors into components solves most problems
- Know dot vs cross: One gives a scalar, the other a vector, never mix them
- Revise from the PDF: Skim it before every physics test and mock
Get the Complete Formula Sheet PDF Free
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