**Chapter: Vectors (Class 12 CBSE Mathematics)**

**Definition of Physical Quantity:**

- A physical quantity is a characteristic or property of an object that can be measured. It typically has a numerical value and a unit.
- Physical quantities are used to describe various aspects of the physical world.

**Types of Physical Quantities:**

**Scalar Quantity:**- Scalar quantities are those that have only magnitude (size) and no direction.
- They are represented by real numbers and units.
- Examples: Mass, Speed, Temperature, Time, Distance.

**Vector Quantity:**- Vector quantities are those that have both magnitude and direction.
- They are represented by arrows or directed line segments.
- Examples: Displacement, Velocity, Force, Acceleration, Momentum.

**Vector Diagram:**

- A vector is represented graphically by an arrow. The length of the arrow represents the magnitude of the vector, and the direction of the arrow indicates the direction of the vector.

**Mathematical Definition of Vectors:**

- Vectors are mathematical entities represented by ordered pairs or triplets in coordinate systems.
- In a two-dimensional (2D) coordinate system, vectors are represented as (x, y), where 'x' is the horizontal component, and 'y' is the vertical component.
- In a three-dimensional (3D) coordinate system, vectors are represented as (x, y, z), where 'x' is the component along the x-axis, 'y' along the y-axis, and 'z' along the z-axis.

**Vector Addition Rule: Triangle Rule and Parallelogram Rule**

**Triangle Rule**:- Graphical method for adding vectors.
- Start with the first vector (vector A) as a straight arrow from the origin.
- Draw the second vector (vector B) starting from the end of the first vector.
- The sum of vectors A and B is represented by the vector pointing from the origin to the end point of vector B.
- Visualization is based on the idea that adding vectors is akin to forming a triangle.
- Suitable for adding two vectors.

**Parallelogram Rule**:- Another graphical method for vector addition.
- Draw two vectors (A and B) as arrows with the same origin.
- Create a parallelogram with these vectors as adjacent sides.
- The diagonal of the parallelogram, starting from the common origin, represents the sum of vectors A and B.
- This method is based on the concept that vectors can be added geometrically by constructing a parallelogram.
- Useful when adding two vectors or more than two vectors.

Both the Triangle Rule and Parallelogram Rule help visualize vector addition. They allow you to determine the magnitude and direction of the resultant vector when adding two or more vectors. The choice between these methods depends on the number and arrangement of vectors you're working with.

**Scalar Multiplication of Vectors:**

- Scalar multiplication of a vector involves multiplying the vector by a scalar (a real number).
- This operation changes the magnitude of the vector but leaves its direction unchanged.

**Vector Subtraction:**

- Vector subtraction can be done by adding the negative of a vector to another vector.
- For vector A and its negative -A, A - (-A) = A + A = 2A.

**Unit Vector:**

- A unit vector is a vector with a magnitude of 1 and is often used to indicate direction.
- It is represented as "û."
- Example: In 2D, the unit vector along the x-axis is ûᵢ = (1, 0), and along the y-axis is ûⱼ = (0, 1).

**Scalar and Vector Products:**

- The dot product (scalar product) and cross product (vector product) are two fundamental operations involving vectors.
- Dot Product: A·B = |A||B|cosθ, where θ is the angle between A and B.
- Cross Product: A × B = |A||B|sinθn, where θ is the angle between A and B, and 'n' is a unit vector perpendicular to the plane formed by A and B.

**Vector Geometry:**

- Vectors can be used to study geometry, solve problems related to lines and planes, and determine angles and distances.
- They are essential in various fields, including physics, engineering, computer graphics, and navigation.

Understanding vectors and their mathematical properties is crucial for solving problems in physics, engineering, and other sciences. They provide a powerful tool for describing and analyzing physical phenomena and geometric relationships.

**Position Vector:**

- A position vector, often denoted as 'r,' represents the location of a point in space with respect to a reference point (origin). It points from the origin to the given point.
- In three-dimensional space (3D), a position vector is typically represented as r = (x, y, z), where 'x,' 'y,' and 'z' are the coordinates of the point.

**Direction Angle (θ) and Direction Ratios:**

- Direction angle (θ) is the angle between a vector and a specified axis (usually the positive x-axis). It measures the orientation of the vector in space.
- Direction ratios are the ratios of the components of a vector along the coordinate axes. For vector A = (a, b, c), the direction ratios are a, b, and c.

**Direction Cosines:**

- Direction cosines (l, m, n) are the cosines of the angles between a vector and each of the coordinate axes (x, y, z).
- Mathematically, they are defined as: l = cos(α), m = cos(β), n = cos(γ), where α, β, and γ are the angles between the vector and the x, y, and z-axes, respectively.

**Mathematical Relations:**

- Direction cosines are related by the equation: l² + m² + n² = 1. This relation ensures that the direction cosines form a unit vector.

**Unit Vector:**

- A unit vector, denoted as û, has a magnitude of 1 and is used to represent the direction of a vector. It is often used to describe the orientation of a vector without considering its magnitude.
- Mathematically, a unit vector in the direction of a vector A is given by: û = (l, m, n), where l, m, and n are the direction cosines of vector A.
- To obtain the unit vector, divide each component of the vector by its magnitude: û = A / |A|, where |A| represents the magnitude of vector A.

**Components of a Vector in 3-D:**

- In 3D space, a vector A can be represented as A = (A₁, A₂, A₃), where A₁, A₂, and A₃ are its components along the x, y, and z-axes, respectively.
- A vector A can be decomposed into its components using its direction cosines: A₁ = |A| * l, A₂ = |A| * m, A₃ = |A| * n.

**Vector Representation in Components (Examples):**

- Vector A = (3, -2, 5)
- Vector B = (1, 1, 1)
- Vector C = (-4, 0, 0)
- Vector D = (0, 5, -3)
- Vector E = (2√2, -√2, 3)

**Dot Product Explanation:**

- The dot product (scalar product) of two vectors A and B is defined as A · B = |A| |B| cos(θ), where θ is the angle between A and B.
- The dot product is commutative: A · B = B · A.
- It is distributive over addition: A · (B + C) = A · B + A · C.
- The dot product is used to find the angle between two vectors and project one vector onto another.

**Vector Product (Cross Product) Explanation:**

- The vector product (cross product) of two vectors A and B is denoted as A × B.
- Its magnitude is |A × B| = |A| |B| sin(θ), where θ is the angle between A and B.
- The direction of A × B is perpendicular to both A and B, following the right-hand rule.
- The cross product is used to find vectors perpendicular to a plane defined by two vectors.

**Numerical Problems (10):**

- Calculate the dot product of vectors A = (3, -2, 5) and B = (1, 4, -2).
- Find the angle between vectors P = (2, -3, 1) and Q = (-1, 5, -4).
- Compute the cross product of vectors X = (2, 3, -1) and Y = (-1, -2, 3).
- Determine a unit vector in the direction of vector M = (4, -1, 2).
- Express vector V = (6, -8, 10) in terms of its unit vector and magnitude.
- Given the direction cosines l = 2/3, m = -1/3, and n = 2/3, find the angle θ that vector A makes with the x-axis.
- If the dot product of vectors A and B is 10 and the magnitude of A is 5, find the magnitude of vector B.
- Calculate the component of vector C = (3, 2, -1) along the x-axis.
- Compute the cross product of vectors U = (1, 2, -3) and V = (2, -1, 4).
- Given vectors X = (3, -2, 1) and Y = (1, 2, -2), find the vector X × Y and its magnitude.