Vectors PDF

# Chapter 1: Vectors ## Contents - 1.1 Introduction - 1.2 Definition of a vector - 1.3 Types of vectors - 1.3.1 Zero vector - 1.3.2 Equal vectors - 1.3.3 Negative of a vector - 1.3.4 Unit vector - 1.4 Components of a vector - 1.5 Magnitude of a vector - 1.5.1 Properties of magn...

# Chapter 1: Vectors ## Contents - 1.1 Introduction - 1.2 Definition of a vector - 1.3 Types of vectors - 1.3.1 Zero vector - 1.3.2 Equal vectors - 1.3.3 Negative of a vector - 1.3.4 Unit vector - 1.4 Components of a vector - 1.5 Magnitude of a vector - 1.5.1 Properties of magnitude - 1.6 Operations on vectors - 1.6.1 Vector addition and subtraction - a) Analytical method - b) Geometric methods - c) Properties of vector addition - 1.6.2 Multiplying vectors by a scalar - a) Properties of scalar multiplication - b) Collinear Vectors - 1.7 Scalar product - 1.7.1 Geometric definition - 1.7.2 Cartesian expression - 1.7.3 Properties of the scalar product - 1.7.4 The angle between two vectors - 1.8 Vector product - 1.8.1 Geometric definition - 1.8.2 Cartesian expression - 1.8.3 Properties of the cross product - 1.9 Mixed product - 1.9.1 Properties of mixed product - 1.10 Derivatives of vectors - 1.10.1 Time derivatives of rotating unit vectors ## 1.1 Introduction In physics, several quantities are completely determined by a real number to which a unit is associated, such as time, mass, volume, etc. We are talking about a displacement lasting 30 seconds, of an object with a mass of 80 kg; these quantities are called scalars. However, certain physical quantities cannot be solely associated with a real number; it may be necessary to specify a spatial orientation to fully define this quantity. In this case, we refer to it as a vector quantity and represent it with a vector. ## 1.2 Definition of a vector A vector is a specific quantity that has a magnitude and direction. It is represented by an arrow and is labeled with lowercase or uppercase letters, with a small arrowhead on top. For example: _w_, _V_, _AB_, etc.. The magnitude of a vector represents its length and the direction of a vector is from its tail (initial point) to its head (terminal point) (see Figure 1.1). ## 1.3 Types of vectors ### 1.3.1 Zero vector The zero vector is a vector that has a magnitude equal to 0. It has no direction and is denoted as _0_. ### 1.3.2 Equal vectors Two vectors _v_ and _w_ are considered equal if they have the same magnitude (length) and the same direction. Given that a vector is represented by an arrow, any translation of this arrow represents the same vector. ### 1.3.3 Negative of a vector The vector _v_ is said to be the negative of vector _w_ if they have the same magnitude but are opposite in direction. This is denoted as: _v_ = −_w_. ### 1.3.4 Unit vector A vector _û_ for which the magnitude is equal to 1 unit length is called a unit vector and denoted by _û_ cap or hat: _û_. **Theorem** For any non-zero vector _v_, the vector **v**^ = _v_ / ||_v_|| is a unit vector that has the same direction as _v_. ## 1.4 Components of a vector **Theorem** Let _v_ be a vector defined in a three-dimensional space. The space is referred to the coordinate system (0,i,j,k). There exists a unique ordered triplet of real numbers (x;y;z) so that: _v_ = x î + y ĵ + zk . The real numbers x, y and z are the coordinates or components of the vector _v_ in terms of the coordinate system (0,i,j,k). x represents the abscissa, y denotes the ordinate, and z represents the altitude of the vector _v_. The vector _v_ shown in Figure 1.2 can be written in terms of its components: _v_ = v_x_ + v_y_ + v_z_ = x î + y ĵ + zk. ## 1.5 Magnitude of a vector The magnitude (length) of a vector _v_ = x î + y ĵ + zk is a non-negative scalar quantity represented by either double vertical lines ||_v_||, an absolute value |_v_|, or simply _v_. It is calculated based on its components by the Pythagorean theorem as: ||_v_|| = √x² + y² + z². **Example 1.1** Consider the given vectors _v_ and _w_, which are defined as follows: _v_ = 3 î + 4 k and _w_ = −3 î – 4 ĵ. Calculate ||_v_|| and ||_w_||. **Solution** ||_v_|| = √3² + 0² + 4² = 5 ||_w_|| = √(-3)² + (-4)² + 0² = 5 ## 1.5.1 Properties of magnitude Since the magnitude is defined as the length of the vector, it must possess the following properties: Let _v_ and _w_ be two vectors and _λ_ be a real number, then: 1. ||_v_|| ≥ 0; 2. ||_v_|| = 0, if and only if _v_ = 0 ; 3. ||−_v_|| = ||_v_|| ; 4. ||_λ_ v|| = |_λ_ | ||_v_|| ; 5. ||_v_ + _w_|| ≤ ||_v_|| + ||_w_|| ## 1.6 Operations on vectors ### 1.6.1 Vector addition and subtraction #### a) Analytical method Given two vectors _A_ = x î + y ĵ + zk and _B_ = x' î + y' ĵ + z' k. The sum of these two vectors is the vector _C_ noted: _C_ = _A_ + _B_ = (x + x') î + (y + y') ĵ + (z + z') k. **Example 1.2** Using the vectors provided in Example 1.1, find: ||_v_|| + ||_w_|| and ||_v_ + _w_|| **Solution** ||_v_|| + ||_w_|| = 5 + 5 = 10. _v_ + _w_ = −4 ĵ + 4 k, so: || _v_ + _w_ || = √32. #### b) Geometric methods We can graphically obtain the sum _R_ (the resultant vector) using what is called the Chasles' relation. It consists of placing the vectors _v_ and _w_ in such a way that the tail of _w_ coincides with the head of _v_. The resultant _R_ is the vector whose tail is the same as the first vector (point A) and its head is the same as the last vector (point C). This method is called the Triangle law. We can also use another geometric method known as the Parallelogram law. This method consists of placing the tails of the two vectors to be added (_v_ and _w_ in our example) at the same point. The resultant vector _R_ is obtained from the diagonal of the parallelogram formed by _v_ and _w_ (see Figure 1.3). #### b) Properties of vector addition Suppose the vectors _u_, _v_, and _w_ are defined in a three-dimensional space, we have: 1. Associativity: (_u_ + _v_) + _w_ = _u_ + (_v_ + _w_). 2. Commutativity: _v_ + _w_ = _w_ + _v_. 3. Additive identity: This is the zero vector. We have: _v_ + 0 = 0 + _v_ = _v_. 4. Additive inverse: −_v_ is the opposite vector of vector _v_. So: _v_ + (−_v_) = 0. The subtraction of the two vectors _v_ and _w_ can be defined as the addition of the two vectors _v_ and –_w_, where –_w_ represents the negative vector of _w_. Then, we write: _v_ - _w_ = _v_ + (−_w_). ### 1.6.2 Multiplying vectors by a scalar Given a vector _w_ with (_x_, _y_, _z_) components, and _λ_ a scalar (real number). The multiplication of _λ_ by _w_ is a vector _B_ defined as follows: _B_ = _λ_ _w_ = (_λ_x) î + (_λ_y) ĵ + (_λ_z) k. - If ||_w_|| = _a_, then ||_B_|| = |_λ_ |a, - If _λ_ is positive, _B_ keeps the same direction as _w_, - If _λ_ is negative, the direction of _B_ is reversed (opposite direction of _w_), - _B_ is a zero vector, if _x_ = 0 or if _w_ is a zero vector. #### a) Properties of scalar multiplication Let _v_ and _w_ be two vectors, and let _λ_ and _μ_ be two real numbers. Then: 1. _λ_ (_v_ + _w_) = _λ_ _v_ + _λ_ _w_ ; 2. (_λ_ + _μ_) _v_ = _λ_ _v_ + _μ_ v_; 3. _λ_ (_μ_ _v_) = (_λ_ _μ_) _v_. #### b) Collinear Vectors Let _v_ and _w_ be two non-zero vectors, we say that _v_ and _w_ are collinear if there exists _λ_ ∈ *R* such that: _v_ = _λ_ _w_. In other words, two vectors are collinear if one is a scalar multiple of the other. ## 1.7 Scalar product The scalar product (also known as the dot product or the inner product) of two vectors _A_ and _B_ is an algebraic operation denoted as _A_ ⋅ _B_ (read as _A_ dot _B_) which takes two vectors as arguments and returns a scalar as a result (hence its name). ### 1.7.1 Geometric definition The dot product of two non-zero vectors _A_ and _B_ is given by the geometric definition as follows: _A_ ⋅ _B_ = ||_A_|| ||_B_|| cos _θ_ where _θ_ (0 ≤ _θ_ ≤ π) represents the angle between _A_ and _B_. If one of the two vectors is null (zero), then the dot product is null. **Example 1.3** Suppose 2 and 3 are the magnitudes of the two vectors _u_ and _v_ respectively, which form an angle of 180° between them. Find _u_ ⋅ _v_. **Solution** _u_ ⋅ _v_ = 2 × 3 cos 180 = -6 ### 1.7.2 Cartesian expression In the context of an orthonormal coordinate system (0,i,j,k), the dot product of two vectors _A_ and _B_ can be expressed in terms of their respective components (_x_, _y_, _z_) and (_x'_, _y'_, _z'_). This is defined by the following relation: _A_ ⋅ _B_ = _xx′_ + _yy′_ + _zz′_ **Example 1.4** Consider the vectors, _u_ and _v_ such that: _u_ = 3 î + 2 ĵ and _v_ = î − ĵ + k. Compute _u_ ⋅ _v_. **Solution** _u_ ⋅ _v_ = (3 × 1) + (2 × − 1) + (0 × 1) = 1. ### 1.7.3 Properties of the scalar product For any vectors _v_ and _w_ and any scalar _λ_ ∈ _R_, we have: 1. Commutativity: _v_ ⋅ _w_ = _w_ ⋅ _v_ ; 2. Distributivity: _u_ ⋅ (_v_ + _w_) = _u_ ⋅ _v_ + _u_ ⋅ _w_ ; 3. Orthogonality: If _v_⊥_w_, then _v_ ⋅ _w_ = 0 ; 4. (_λ_ _v_) ⋅ _w_ = _λ_ (_v_ ⋅ _w_) ; 5. _v_ ⋅ _v_ = ||_v_||²; 6. _v_ ⊥ 0 = 0. ### 1.7.4 The angle between two vectors The dot product allows, among other things, to measure the angle between two vectors. **Theorem** Let _A_ and _B_ be two non-zero vectors, the angle _θ_, 0 ≤ _θ_ ≤ _π_, between the vectors _A_ and _B_ is given by the formula: _θ_ = arccos (_A_ ⋅ _B_) / (||_A_|| ||_B_||) _θ_ = arccos [_A_ ⋅ _B_] / [||_A_|| ||_B_||] give the acute angle. **Example 1.5** Find the angle _α_ between the two vectors _u_ and _v_ from Example 1.4. **Solution** cos _α_ = (_u_ ⋅ _v_) / (||_u_|| ||_v_||) = 1 / (√13 √3) = 0.18 ⇒ _α_ = 79.18° ## 1.8 Vector product The vector product (also known as the cross product or the outer product) of two vectors _A_ and _B_ is a vector operation, denoted as _A_ × _B_ (read as _A_ cross _B_), performed in space (the cross product does not exist in two-dimensions) that takes two vectors as arguments and returns a vector as a result (hence its name). ### 1.8.1 Geometric definition Let _v_ = _OA_ and _w_ = _OB_ be two vectors in an orthonormal coordinate system (0,i,j,k), forming an angle _α_ between them. By definition, the cross product _v_ × _w_ is the vector _a_ whose magnitude is equal to: ||_a_|| = ||_v_|| ||_w_|| sin _α_, and its direction is perpendicular to the plane formed by _v_ and _w_ (see Figure 1.4). This is given by the right-hand rule (the right-handed screw rotated from _v_ to _w_). The area of parallelogram _OACB_ is: S_OACB_ = _OB_ ⋅ _hA_ = ||_v_ × _w_|| = ||_a_|| = ||_v_|| ||_w_|| sin _α_. The area of triangl _OAB_ (pounded by vectors _v_ and _w_) is: 1/2 S_OACB_ = 1/2 (_OB_ ⋅ _hA_) = 1/2 ||_v_ × _w_|| = 1/2 ||_a_||. ### 1.8.2 Cartesian expression Let us consider two vectors _v_ = (_x_,_y_,_z_) and _w_ = (_x'_,_y'_,_z'_), whose components are expressed in the orthonormal basis (i,j,k). The cross product _v_ × _w_ can also be defined as: | i j k | |-----| | _x_ _y_ _z_ | | _x'_ _y'_ _z'_ | _v_ × _w_ = (_yz'_- _zy'_) î + (_zx'_- _xz'_) ĵ + (_xy'_- _yx'_) k **Example 1.6** Find _v₁_ × _v₂_, knowing that: _v₁_ = (3√3 / 2, 3 / 2) and _v₂_ = (2, 0). **Solution** _v₁_ × _v₂_ = | i j k | |-----| | 3√3 / 2 3 / 2 0 | | 2 0 0 | = 0 î - 0 ĵ - 3 k = -3 k. ### 1.8.3 Properties of the cross product For vectors _u_, _v_ and _w_ and scalars _λ_ and _μ_, we have: 1. The cross product is not commutative: _v_ × _w_ = −_w_ × _v_ ; 2. The cross product is distributive over vector sum: _u_ × (_v_ + _w_) = _u_ × _v_ + _u_ × _w_ ; 3. (_λ_ _v_) × (_μ_ _w_) = _λ_ _μ_ (_v_ × _w_) ; 4. (_λ_ _v_) × _w_ = _λ_ (_v_ × _w_) = _v_ × (_λ_ _w_) ; 5. _v_ and _w_ are colinear if and only if _v_ × _w_ = 0. ## 1.9 Mixed product In a coordinate system (O,i,j,k), the scalar triple product of three vectors _a_, _b_, and _c_, taken in that order, is the real number (scalar quantity) denoted as [_a_,_b_,_c_] and defined by: [_a_,_b_,_c_] = _a_ ⋅ (_b_ × _c_). In three-dimensional space, the components of vectors _a_,_b_, and _c_ are denoted as (_a₁_,_a₂_, _a₃_), (_b₁_,_b₂_,_b₃_), and (_c₁_,_c₂_,_c₃_), respectively. The scalar triple product of these vectors can be expressed as a determinant involving their components, | _a₁_ _b₁_ _c₁_ | |-----| | _a₂_ _b₂_ _c₂_ | | _a₃_ _b₃_ _c₃_ | [_a_,_b_,_c_] = _a₁_ | _b₂_ _c₃_ | |-----| | _b₃_ _c₂_ | - _a₂_ | _b₁_ _c₃_ | |-----| | _b₃_ _c₁_ | + _a₃_ | _b₁_ _c₂_ | |-----| | _b₂_ _c₁_ | = _a₁_ (_b₂_ _c₃_ - _b₃_ _c₂_) - _a₂_ (_b₁_ _c₃_ - _b₃_ _c₁_) + _a₃_ (_b₁_ _c₂_ -_b₂_ _c₁_) = | _a₁_ _b₁_ _c₁_ | |-----| | _a₂_ _b₂_ _c₂_ | | _a₃_ _b₃_ _c₃_ | ### 1.9.1 Properties of mixed product 1. The scalar triple product changes sign when two vectors are swapped: [_a_,_b_,_c_] = − [_b_,_a_,_c_] = − [_c_,_b_,_a_] = −[_a_,_c_,_b_]; 2. The scalar triple product is invariant under a circular permutations of its vectors [_a_,_b_,_c_] = [_b_,_c_,_a_] = [_c_,_a_,_b_]; 3. (_a_ × _b_) ⋅ _c_ = _a_ ⋅ (_b_ × _c_); 4. [_a_, _b_, _c_ ] = [_a_, _b_, _c_], ∀_λ_ ∈ *R*; 5. [_a_,_b_,_c_ + _d_]= [_a_,_b_,_c_]+[_a_,_b_,_d_]; 6. [_a_,_b_,_c_] represents the volume of the parallelepiped with coterminous edges _a_, _b_, and _c_. **Example 1.7** Let there be a parallelepiped generated by three vectors _OA_, _OB_ and _OC_. Demonstrate that its volume is equal to the scalar triple product of the vectors _OA_, _OB_ and _OC_. ## 1.10 Derivatives of vectors If _x_, _y_ and _z_ are the components of a vector _v_ expressed as a function to time. The derivative of the vector _v_ with respect to time is a vector, denoted _dv_/dt or _v_′, which has the following components: _dx_/dt, _dy_/dt and _dz_/dt Then: _dv_/dt= (_dx_/dt) î + (_dy_/dt) ĵ + (_dz_/dt) k. The second derivative of _v_ with respect to time can be written as: _d²v_/dt² = _v_″ = _v_′ = (_d²x_/dt²) î + (_d²y_/dt²) ĵ + (_d²z_/dt²) k. ### 1.10.1 Time derivatives of rotating unit vectors Given _û_ and _ñ_ two orthogonal unit vectors represented in the Cartesian base system (O,i,j,k). _û_ and _ñ_ can rotate around the _Oz_ axis with the same angle _θ_ (see Figure 1.6). We have: _û_ = cos _θ_ î + sin _θ_ ĵ and _ñ_ = − sin _θ_ î + cos _θ_ ĵ. _dû_/dt = _dû_/d_θ_ * d_θ_/dt = (d/d_θ_ (−sin _θ_ î + cos _θ_ ĵ) d_θ_/dt = (−cos _θ_ î + sin _θ_ ĵ) d_θ_/dt _dû_/dt = - _θ'_ _ñ_ Similarly, we find: _dñ_/dt = _θ'_ _û_ We can therefore generalize this result for any rotating unit vector, where _θ_ = _ω_ represents the angular velocity.