Class 11 Physics: Vectors

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### Introduction to Vectors - **Scalar Quantity:** A physical quantity described by magnitude only (e.g., mass, distance, speed, time, temperature, energy). - **Vector Quantity:** A physical quantity described by both magnitude and direction (e.g., displacement, velocity, acceleration, force, momentum). - **Representation:** A vector is represented by a directed line segment. The length of the segment represents its magnitude, and the arrow indicates its direction. - Notation: $\vec{A}$ or $\mathbf{A}$ - Magnitude: $|\vec{A}|$ or $A$ ### Types of Vectors - **Null Vector (Zero Vector):** A vector with zero magnitude and arbitrary direction. $\vec{0}$. - **Unit Vector:** A vector with magnitude 1. Used to specify direction. - $\hat{A} = \frac{\vec{A}}{|\vec{A}|}$ - **Co-initial Vectors:** Vectors having the same initial point. - **Collinear Vectors:** Vectors acting along the same line or parallel lines. - Can be in the same direction (parallel) or opposite direction (anti-parallel). - **Coplanar Vectors:** Vectors lying in the same plane. - **Equal Vectors:** Two vectors are equal if they have the same magnitude and same direction. - **Negative Vector:** A vector having the same magnitude as the given vector but acting in the opposite direction. $\vec{B} = -\vec{A}$ - **Position Vector:** A vector that specifies the position of a point with respect to the origin of a coordinate system. $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ - **Displacement Vector:** The vector from the initial position to the final position. - If initial position is $P_1(x_1, y_1, z_1)$ and final position is $P_2(x_2, y_2, z_2)$, then $\vec{P_1P_2} = (x_2-x_1)\hat{i} + (y_2-y_1)\hat{j} + (z_2-z_1)\hat{k}$ ### Vector Addition - **Triangle Law of Vector Addition:** 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 taken in the opposite order. - If $\vec{A}$ and $\vec{B}$ are two vectors, then $\vec{R} = \vec{A} + \vec{B}$. - Magnitude: $|\vec{R}| = \sqrt{A^2 + B^2 + 2AB\cos\theta}$, where $\theta$ is the angle between $\vec{A}$ and $\vec{B}$. - Direction: $\tan\alpha = \frac{B\sin\theta}{A+B\cos\theta}$, where $\alpha$ is the angle of $\vec{R}$ with $\vec{A}$. - **Parallelogram Law of Vector Addition:** If two vectors originating from the same point are represented by the two adjacent sides of a parallelogram, then their resultant is represented by the diagonal passing through that common point. - Same formulas for magnitude and direction as Triangle Law. - **Polygon Law of Vector Addition:** If a number of vectors are represented by the sides of an open polygon taken in the same order, then their resultant is represented by the closing side of the polygon taken in the opposite order. - $\vec{R} = \vec{A} + \vec{B} + \vec{C} + ...$ - **Properties of Vector Addition:** - **Commutative:** $\vec{A} + \vec{B} = \vec{B} + \vec{A}$ - **Associative:** $(\vec{A} + \vec{B}) + \vec{C} = \vec{A} + (\vec{B} + \vec{C})$ - **Identity:** $\vec{A} + \vec{0} = \vec{A}$ - **Inverse:** $\vec{A} + (-\vec{A}) = \vec{0}$ ### Vector Subtraction - Subtraction of vector $\vec{B}$ from $\vec{A}$ is defined as the addition of vector $-\vec{B}$ to $\vec{A}$. - $\vec{R} = \vec{A} - \vec{B} = \vec{A} + (-\vec{B})$ - Magnitude: $|\vec{R}| = \sqrt{A^2 + B^2 - 2AB\cos\theta}$ - Direction: $\tan\alpha = \frac{B\sin\theta}{A-B\cos\theta}$ (angle with $\vec{A}$) ### Resolution of Vectors - The process of splitting a vector into two or more components. - **Rectangular Components (2D):** A vector $\vec{A}$ making an angle $\theta$ with the x-axis can be resolved into: - X-component: $A_x = A\cos\theta$ - Y-component: $A_y = A\sin\theta$ - $\vec{A} = A_x\hat{i} + A_y\hat{j}$ - Magnitude: $A = \sqrt{A_x^2 + A_y^2}$ - Direction: $\tan\theta = \frac{A_y}{A_x}$ - **Rectangular Components (3D):** $\vec{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k}$ - Magnitude: $A = \sqrt{A_x^2 + A_y^2 + A_z^2}$ - Direction cosines: $\cos\alpha = \frac{A_x}{A}$, $\cos\beta = \frac{A_y}{A}$, $\cos\gamma = \frac{A_z}{A}$ - $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$ ### Multiplication of Vectors #### 1. Scalar (Dot) Product - The scalar product of two vectors $\vec{A}$ and $\vec{B}$ is a scalar quantity. - $\vec{A} \cdot \vec{B} = AB\cos\theta$ - **Properties:** - Commutative: $\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}$ - Distributive: $\vec{A} \cdot (\vec{B} + \vec{C}) = \vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{C}$ - $\hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1$ - $\hat{i} \cdot \hat{j} = \hat{j} \cdot \hat{k} = \hat{k} \cdot \hat{i} = 0$ - If $\vec{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k}$ and $\vec{B} = B_x\hat{i} + B_y\hat{j} + B_z\hat{k}$ Then $\vec{A} \cdot \vec{B} = A_xB_x + A_yB_y + A_zB_z$ - If $\vec{A} \cdot \vec{B} = 0$ and $\vec{A}, \vec{B} \neq \vec{0}$, then $\vec{A}$ is perpendicular to $\vec{B}$. - Physical examples: Work done ($W = \vec{F} \cdot \vec{d}$), Power ($P = \vec{F} \cdot \vec{v}$) #### 2. Vector (Cross) Product - The vector product of two vectors $\vec{A}$ and $\vec{B}$ is a vector quantity. - $\vec{A} \times \vec{B} = (AB\sin\theta)\hat{n}$, where $\hat{n}$ is a unit vector perpendicular to the plane containing $\vec{A}$ and $\vec{B}$, given by the right-hand thumb rule. - Magnitude: $|\vec{A} \times \vec{B}| = AB\sin\theta$ - **Properties:** - Not Commutative: $\vec{A} \times \vec{B} = -(\vec{B} \times \vec{A})$ - Distributive: $\vec{A} \times (\vec{B} + \vec{C}) = \vec{A} \times \vec{B} + \vec{A} \times \vec{C}$ - $\hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = \vec{0}$ - $\hat{i} \times \hat{j} = \hat{k}$, $\hat{j} \times \hat{k} = \hat{i}$, $\hat{k} \times \hat{i} = \hat{j}$ - $\hat{j} \times \hat{i} = -\hat{k}$, $\hat{k} \times \hat{j} = -\hat{i}$, $\hat{i} \times \hat{k} = -\hat{j}$ - If $\vec{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k}$ and $\vec{B} = B_x\hat{i} + B_y\hat{j} + B_z\hat{k}$ Then $\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix} = (A_yB_z - A_zB_y)\hat{i} - (A_xB_z - A_zB_x)\hat{j} + (A_xB_y - A_yB_x)\hat{k}$ - If $\vec{A} \times \vec{B} = \vec{0}$ and $\vec{A}, \vec{B} \neq \vec{0}$, then $\vec{A}$ is parallel to $\vec{B}$. - Physical examples: Torque ($\vec{\tau} = \vec{r} \times \vec{F}$), Angular Momentum ($\vec{L} = \vec{r} \times \vec{p}$) ### Applications - **Relative Velocity:** $\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$ (velocity of A with respect to B) - **Projectile Motion:** Vector concepts are fundamental to analyzing projectile trajectories. - **Forces:** All forces are vector quantities. Newton's laws are vector equations.