Class 12 Maths Vector Algebra Notes

10.1 Introduction

Vectors are quantities that have both magnitude and direction, unlike scalars which have only magnitude.

• Widely used in physics, engineering, and computer graphics
• Useful for representing forces, velocities, and displacements
• Vector algebra simplifies geometrical and physical problems in space

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10.2 Some Basic Concepts

• Vector: Quantity with magnitude and direction (e.g., velocity, force)
• Zero vector: Vector with magnitude 0, denoted by $\vec{0}$0
• Position vector: Vector representing the position of a point relative to the origin
• Unit vector: Vector of magnitude 1, often denoted by $\hat{i}, \hat{j}, \hat{k}$i^,j^​,k^

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10.3 Types of Vectors

1. Zero vector: Magnitude = 0, direction undefined
2. Unit vector: Magnitude = 1
3. Equal vectors: Same magnitude and direction
4. Negative vector: Opposite in direction to a given vector
5. Position vector: From origin to a point in space

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10.4 Addition of Vectors

• Triangle Law: Place tail of second vector at head of first; resultant from tail of first to head of second
• Parallelogram Law: Place vectors with same initial point; diagonal of parallelogram gives resultant

$$\vec{A} + \vec{B} = \vec{R}$$A+B=R

• Properties of vector addition:
  • Commutative: $\vec{A} + \vec{B} = \vec{B} + \vec{A}$A+B=B+A
  • Associative: $\vec{A} + (\vec{B} + \vec{C}) = (\vec{A} + \vec{B}) + \vec{C}$A+(B+C)=(A+B)+C

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10.5 Multiplication of a Vector by a Scalar

• Multiplying vector $\vec{A}$A by scalar $k$k:

$$k \vec{A}$$kA

• Magnitude: $|k \vec{A}| = |k| |\vec{A}|$∣kA∣=∣k∣∣A∣
• Direction: Same as $\vec{A}$A if k > 0, opposite if k < 0

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10.6 Product of Two Vectors

🔹 1. Scalar (Dot) Product

• For vectors $\vec{A}$A and $\vec{B}$B:

$$\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta$$A⋅B=∣A∣∣B∣cosθ

• Results in a scalar
• Properties:
  • Commutative: $\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}$A⋅B=B⋅A
  • Distributive: $\vec{A} \cdot (\vec{B} + \vec{C}) = \vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{C}$A⋅(B+C)=A⋅B+A⋅C

🔹 2. Vector (Cross) Product

• For vectors $\vec{A}$A and $\vec{B}$B:

$$\vec{A} \times \vec{B} = |\vec{A}| |\vec{B}| \sin \theta \, \hat{n}$$A×B=∣A∣∣B∣sinθn^

• Results in a vector perpendicular to both $\vec{A}$A and $\vec{B}$B
• Direction given by right-hand rule

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✅ Key Points to Remember

• Vectors have magnitude and direction; scalars have only magnitude
• Addition is commutative and associative
• Scalar multiplication changes magnitude; may reverse direction
• Dot product gives scalar, cross product gives vector perpendicular to plane
• Vector algebra simplifies geometry and physics problems in 2D and 3D