# Chapter 2: Vectors 📐

## Introduction to Scalars and Vectors

Physical quantities can be broadly classified into two categories: scalars and vectors.

* [cite_start]**Scalar Quantity:** These are physical quantities that have **magnitude only** and no direction[cite: 2, 3].
    * *Examples:* Mass, length, time, speed, temperature.
* [cite_start]**Vector Quantity:** These are physical quantities that possess both **magnitude and direction**[cite: 5].
    * *Examples:* Velocity, acceleration, force, displacement, momentum.

---

## Types of Vectors

[cite_start]There are several types of vectors used to describe different physical situations[cite: 6].

### 1. Unit Vector ($\hat{A}$)
[cite_start]A vector with a magnitude of one (unity) is called a **unit vector**[cite: 8]. It is used to specify a particular direction.
The unit vector of a vector $\vec{A}$ is given by:
$$ \hat{A} = \frac{\vec{A}}{|\vec{A}|} $$
[cite_start]Where $|\vec{A}|$ is the magnitude of vector $\vec{A}$[cite: 8].

### 2. Null Vector (or Zero Vector)
[cite_start]A vector that has **zero magnitude** but a specific direction is called a null vector[cite: 11]. [cite_start]Its initial and terminal points are the same[cite: 12].

### 3. Co-initial Vectors
[cite_start]Vectors that share the **same starting point** (initial point) are known as co-initial vectors[cite: 14, 15].

### 4. Co-terminal Vectors
[cite_start]Vectors whose **end points coincide** are called co-terminal vectors[cite: 20].

### 5. Equal Vectors
[cite_start]Two or more vectors are considered equal if they have the **same magnitude and the same direction**, regardless of their initial positions[cite: 26].

### 6. Negative Vector
[cite_start]The negative of a vector has the **same magnitude** but the **opposite direction** to the original vector[cite: 34, 35]. If $\vec{A}$ is a vector, its negative vector is $-\vec{A}$.

### 7. Coplanar Vectors
[cite_start]A set of vectors that all lie on the **same plane** are called coplanar vectors[cite: 38, 39].

### 8. Collinear Vectors
[cite_start]Vectors that act along the **same line** or along parallel or anti-parallel lines are called collinear vectors[cite: 42, 43].

### 9. Like and Unlike Vectors
* [cite_start]**Like Vectors:** Vectors having the same direction[cite: 48].
* [cite_start]**Unlike Vectors:** Vectors having opposite directions[cite: 48].

### 10. Position Vector ($\vec{r}$)
[cite_start]A vector that specifies the position of a point with respect to an origin is called a position vector[cite: 55]. [cite_start]The line drawn from the origin (O) to the point (P) represents the position vector $\vec{r}$[cite: 56].

---

## Vector Addition ➕

### Triangle Law of Vector Addition
> [cite_start]This law states that if two vectors are represented by two sides of a triangle taken in the same order, then their resultant is represented by the third side of the triangle taken in the opposite order[cite: 67].

Let $\vec{A}$ and $\vec{B}$ be two vectors with an angle $\theta$ between them.

* **Magnitude of Resultant ($\vec{R}$):**
    [cite_start]$$ R = \sqrt{A^2 + B^2 + 2AB\cos\theta} $$ [cite: 91]
* **Direction of Resultant ($\vec{R}$):**
    If the resultant $\vec{R}$ makes an angle $\beta$ with vector $\vec{A}$, then:
    [cite_start]$$ \beta = \tan^{-1}\left(\frac{B\sin\theta}{A + B\cos\theta}\right) $$ [cite: 102]

### Parallelogram Law of Vector Addition
> [cite_start]This law states that if two vectors acting at a point are represented by the two adjacent sides of a parallelogram, then the diagonal of the parallelogram passing through that point represents their resultant vector in both magnitude and direction[cite: 106].

[cite_start]The formulas for the magnitude and direction of the resultant are the same as in the Triangle Law[cite: 136, 148].

### Special Cases for Vector Addition
1.  **When vectors are parallel ($\theta = 0^\circ$):**
    * [cite_start]$R = A + B$ (Maximum resultant) [cite: 154]
    * [cite_start]$\beta = 0^\circ$ [cite: 156]
2.  **When vectors are perpendicular ($\theta = 90^\circ$):**
    * [cite_start]$R = \sqrt{A^2 + B^2}$ [cite: 162]
    * [cite_start]$\beta = \tan^{-1}\left(\frac{B}{A}\right)$ [cite: 163]
3.  **When vectors are anti-parallel ($\theta = 180^\circ$):**
    * [cite_start]$R = |A - B|$ (Minimum resultant) [cite: 169]
    * [cite_start]$\beta = 0^\circ$ or $180^\circ$ [cite: 171]

### Polygon Law of Vector Addition
> [cite_start]This law states that if a number of vectors can be represented by the sides of a polygon taken in the same order, then their resultant is represented by the closing side of the polygon taken in the opposite order[cite: 173, 174].
[cite_start]Essentially, $\vec{R} = \vec{A} + \vec{B} + \vec{C} + \vec{D}$[cite: 208].

---

## Resolution of Vectors

[cite_start]The process of splitting a single vector into two or more mutually perpendicular components is known as the **resolution of a vector**[cite: 225].
For a vector $\vec{A}$ making an angle $\theta$ with the x-axis:

* [cite_start]**Horizontal Component ($A_x$):** $A_x = A\cos\theta$ [cite: 239]
* [cite_start]**Vertical Component ($A_y$):** $A_y = A\sin\theta$ [cite: 243]

The vector can be represented in terms of its components and unit vectors ($\hat{i}, \hat{j}, \hat{k}$):
* [cite_start]In 2D: $\vec{A} = A_x\hat{i} + A_y\hat{j}$ [cite: 272]
* [cite_start]In 3D: $\vec{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k}$ [cite: 283]

The magnitude of the vector can be found from its components:
* [cite_start]In 2D: $A = \sqrt{A_x^2 + A_y^2}$ [cite: 245]
* [cite_start]In 3D: $A = \sqrt{A_x^2 + A_y^2 + A_z^2}$ [cite: 220]

---

## Multiplication of Vectors ✖️

### 1. Scalar Product (Dot Product ⋅)
[cite_start]The product of two vectors that results in a **scalar quantity**[cite: 287].

**Formula:**
[cite_start]$$ \vec{A} \cdot \vec{B} = AB\cos\theta $$ [cite: 289]
[cite_start]Where $\theta$ is the angle between $\vec{A}$ and $\vec{B}$[cite: 288].

**In component form:**
[cite_start]$$ \vec{A} \cdot \vec{B} = A_xB_x + A_yB_y + A_zB_z $$ [cite: 310]

**Properties:**
* [cite_start]It is **commutative**: $\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}$[cite: 291].
* [cite_start]It is **distributive**: $\vec{A} \cdot (\vec{B} + \vec{C}) = \vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{C}$[cite: 294, 296].
* [cite_start]For perpendicular vectors ($\theta = 90^\circ$), $\vec{A} \cdot \vec{B} = 0$[cite: 302].
* [cite_start]For parallel vectors ($\theta = 0^\circ$), $\vec{A} \cdot \vec{B} = AB$[cite: 301].
* [cite_start]For unit vectors: $\hat{i}\cdot\hat{i} = \hat{j}\cdot\hat{j} = \hat{k}\cdot\hat{k} = 1$ and $\hat{i}\cdot\hat{j} = \hat{j}\cdot\hat{k} = \hat{k}\cdot\hat{i} = 0$[cite: 302].

**Examples:**
* [cite_start]Work Done: $W = \vec{F} \cdot \vec{s}$ [cite: 318]
* [cite_start]Power: $P = \vec{F} \cdot \vec{v}$ [cite: 319]

### 2. Vector Product (Cross Product ×)
[cite_start]The product of two vectors that results in a **vector quantity**[cite: 323, 324]. The resultant vector is perpendicular to the plane containing the two original vectors.

**Formula:**
[cite_start]$$ \vec{A} \times \vec{B} = AB\sin\theta\hat{n} $$ [cite: 332]
[cite_start]Where $\hat{n}$ is a unit vector perpendicular to both $\vec{A}$ and $\vec{B}$[cite: 332].

**In determinant form:**
[cite_start]$$ \vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix} $$ [cite: 353]
[cite_start]This expands to: $(A_yB_z - A_zB_y)\hat{i} + (A_zB_x - A_xB_z)\hat{j} + (A_xB_y - A_yB_x)\hat{k}$[cite: 350].

**Properties:**
* [cite_start]It is **not commutative**: $\vec{A} \times \vec{B} = -(\vec{B} \times \vec{A})$[cite: 337].
* [cite_start]It is **distributive**: $\vec{A} \times (\vec{B} + \vec{C}) = \vec{A} \times \vec{B} + \vec{A} \times \vec{C}$[cite: 338].
* [cite_start]For parallel vectors ($\theta = 0^\circ$), $\vec{A} \times \vec{B} = 0$[cite: 344].
* For perpendicular vectors ($\theta = 90^\circ$), $|\vec{A} \times \vec{B}| [cite_start]= AB$[cite: 344].
* [cite_start]For unit vectors: $\hat{i}\times\hat{i} = 0$, and in cyclic order $\hat{i}\times\hat{j} = \hat{k}$, $\hat{j}\times\hat{k} = \hat{i}$, $\hat{k}\times\hat{i} = \hat{j}$[cite: 344].

**Examples:**
* [cite_start]Torque: $\vec{\tau} = \vec{r} \times \vec{F}$ [cite: 365]
* [cite_start]Angular Momentum: $\vec{L} = \vec{r} \times \vec{p}$ [cite: 364]
* [cite_start]Linear Velocity: $\vec{v} = \vec{\omega} \times \vec{r}$ [cite: 363]

---

### Differences Between Scalar and Vector Product

| Feature                 | Scalar (Dot) Product                                                              | Vector (Cross) Product                                                                   |
| ----------------------- | --------------------------------------------------------------------------------- | ---------------------------------------------------------------------------------------- |
| **Result** | [cite_start]Produces a **scalar** quantity[cite: 366].                                        | [cite_start]Produces a **vector** quantity[cite: 366].                                               |
| **Formula** | [cite_start]$\vec{A} \cdot \vec{B} = AB\cos\theta$ [cite: 366]                                 | [cite_start]$\vec{A} \times \vec{B} = AB\sin\theta\hat{n}$ [cite: 366]                                |
| **Commutative Law** | [cite_start]**Obeys** the law: $\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}$[cite: 291].     | [cite_start]**Does not obey** the law: $\vec{A} \times \vec{B} = -(\vec{B} \times \vec{A})$[cite: 366]. |
| **Perpendicular Vectors** | [cite_start]Result is **zero** ($\cos90^\circ = 0$)[cite: 366].                                  | [cite_start]Result is **maximum** in magnitude ($\sin90^\circ = 1$)[cite: 344].                        |
| **Parallel Vectors** | [cite_start]Result is **maximum** in magnitude ($\cos0^\circ = 1$)[cite: 301].                    | [cite_start]Result is **zero** ($\sin0^\circ = 0$)[cite: 366].                                        |