Physics Chp 2 Summary PDF

Physics Chp 2 Rectangular Components of a Vector - Components are the effective parts of a vector in different directions. - In 2D, we have x-component and y-component. - The component along the x-axis is called horizontal or x-component. - The component along the y-axis is called vertical or y-...

Physics Chp 2 Rectangular Components of a Vector - Components are the effective parts of a vector in different directions. - In 2D, we have x-component and y-component. - The component along the x-axis is called horizontal or x-component. - The component along the y-axis is called vertical or y-component. - The x-component and y-components are perpendicular to each other and are called rectangular components. Rectangular Components - Consider a vector 'A' making an angle 'θ' with the horizontal. - The projection OQ of vector A along the x-axis is the x-component (Ax). - The projection OS of vector A along the y-axis is the y-component (Ay). - The vector A in its component form can be written as: A = Ax + Ay Or A = Axi^ + Ayj^ - i and j are the unit vectors representing the x-axis and y-axis directions, respectively. Finding Rectangular Components - We apply trigonometric ratios to find the magnitude of rectangular components Ax and Ay. - Consider the triangle OPQ: sin θ = perpendicular/hypotenuse - If the rectangular components Ax and Ay of a vector A are given, we can find the magnitude 'A' and its direction 'θ' using the following equations: - (𝑂𝑃) = (𝑂𝑄) + (𝑄𝑃) 2 2 2  - (𝑂𝑃) = (𝐴𝑥) + (𝐴𝑦) 2 2 2 - A = √((𝐴𝑥) + (𝐴𝑦) )........(i) 2 2 - To find the direction angle θ: - tan θ = Ay / Ax - θ = 𝑡𝑎𝑛 (Ay / Ax)........(ii) −1 Steps: 1. Use the Pythagorean theorem to find the magnitude 'A' of the vector from its rectangular components Ax and Ay. 2. Apply the trigonometric ratio of tangent to find the direction angle θ of the vector. 3. By using the equations (i) and (i), we can determine the vector's magnitude and direction. 2.2 Product of Vectors - When two vector quantities are multiplied, the product may be a scalar or vector quantity. - The type of product obtained depends on the nature of the given vectors. Scalar Product or Dot Product - When the product of two vector quantities gives a scalar quantity, the product is called a scalar product or dot product. - The scalar product of two vectors A and B is represented by putting a dot (·) between the symbols of the two vectors, denoted as A·B. - Here A and B are the magnitudes of vectors of A and B. Thus the scalar product of these vectors is obtained by multiplying the magnitude by the cosine of the angle between them - The dot product is also known as the "scalar product" or "product". - The dot product between two vectors A and B making an angle θ is defined as: A·B = |A| |B| cos θ ………(i) - Equation (i) can also be written as: A·B = (A cos θ) B………(a) Or A·B = (B cos θ) A………(a) - (A cos θ) is the component of vector A in the direction of vector B. - Similarly, (B cos θ) is the component of vector B in the direction of vector A. Properties of Scalar Product 1. **Commutative Property**: The scalar product of two vectors obeys the commutative property, i.e., A·B = B·A. 2. **Scalar Product of Orthogonal (Perpendicular) Vectors**: The scalar product of two orthogonal (perpendicular) vectors is equal to zero, i.e., A·B = 0. - Similarly, for mutually perpendicular unit vectors, we can show i·j = j·k = k·i = 0. 3. **Scalar Product of Parallel Vectors**: The scalar product of two parallel vectors has a maximum value and is equal to the product of their magnitudes, i.e., A·B = |A| |B| cos 0° = |A| |B|. 4. **Scalar Product of a Vector with Itself**: The scalar product of a vector with itself is equal to the square of its magnitude, i.e., A·A = |𝐴|. 2 5. **Scalar Product of a Unit Vector with Itself**: The dot product of a unit vector with itself is unity, i.e., i·i = j·j = k·k = 1. 6. **Scalar Product with Null Vector**: The dot product of a vector with the null vector is also zero. These properties of the scalar product are important for understanding and applying vector operations in various fields of physics and mathematics. 7. **Scalar Product of Antiparallel Vectors**: The scalar product of two antiparallel vectors is negative, i.e., A·B = -|A| |B| cos 180° = -AB. 8. **Scalar Product in Terms of Rectangular Components**: The scalar product of two vectors in terms of their rectangular components can be given as: A·B = (Ax + Ay + Az) · (Bx + By + Bz) = AxBx + AyBy + AzBz………….(i) = A·B cos θ………………(ii) Vector Product or Cross Product Definition: When the product of two vector quantities gives a vector quantity, the product is called a vector product. Representation: The vector product of two vectors is represented by putting a cross (×) between the symbols of the two vectors, therefore it is also known as the cross product. Angle between Vectors: Let's take two vectors A and B that are making an angle θ with each other. The cross product between these two vectors can be denoted by A×B and is defined as: A×B = |A| |B| sin θ n̂ (2.9) Where n̂ is a unit vector perpendicular to both A and B and the right-hand rule gives the direction of n̂. These properties of the vector product (cross product) are important for understanding and applying vector operations in various fields of physics and mathematics. Definition: The vector product of two vectors A and B is defined as: A × B = |A| |B| sin(θ) n̂ Where: |A| and |B| are the magnitudes of vectors A and B θ is the angle between A and B  n̂ is a unit vector perpendicular to both A and B Direction of the Cross Product: The direction of the cross product (n̂) can be determined using the right-hand rule: Rotate the fingers of your right hand in the direction from the first vector (A) to the second vector (B) through the smaller angle between them. The direction of the erect thumb will give the direction of the cross-product. Examples of Vector Products: * Torque (τ) is an example of a vector product, in which force (F) is multiplied by position vector (r). τ = r × F = r F sin θ * Angular momentum (L) is also an example of a vector product. It is the cross-product of linear momentum P and position vector r, i.e., L = r × P = r p sin θ Properties of the Vector Product: 1. The vector product is not commutative, but anti-commutative: A × B = -B × A 2. The vector product of two parallel or antiparallel vectors is zero: The vector product of a vector with itself is equal to zero. i.e., A x A = AA sin0° = 0 A × B = 0 if A and B are parallel or antiparallel The vector product of a vector with itself is zero: A × A = 0 3. Vector Product of Perpendicular Vectors: The vector product of two perpendicular vectors has a maximum value and is equal to the product of their magnitudes multiplied by the sine of the angle between them. For mutually perpendicular unit vectors, the vector product can be shown as: i × j = k, j × k = i, k × i = j. 4. Vector Product in Terms of Rectangular Components: The vector product of two vectors A and B can be expressed in terms of their rectangular components as: A × B = (Ax i + Ay j + Az k) × (Bx i + By j + Bz k) This can be calculated using the determinant method or the expression: A × B = (AxBy - AyBx) i + (AyBz - AzBy) j + (AzBx - AxBz) k 5. Physical Significance of Vector Product: The vector product gives the area of a plane. If vectors A and B represent adjacent sides of a plane, the magnitude of A × B gives the magnitude of the area (which can be a rectangle or a parallelogram).  The unit vector n̂ gives the direction of the area, which is normal to the plane and can be found using the right-hand rule.