Chapter 1: Vectors PDF

 Chapter 1: Vectors 2-1 Introduction 2-2 Vectors and Scalars 2-3 Adding Vectors Geometrically 2-4 Components of Vectors 2-5 Unit Vector 2-6 Multiplying Vectors 1- Multiplying a vector by a scalar 2- Multiplying a vector by a vector a- Scal...

 Chapter 1: Vectors 2-1 Introduction 2-2 Vectors and Scalars 2-3 Adding Vectors Geometrically 2-4 Components of Vectors 2-5 Unit Vector 2-6 Multiplying Vectors 1- Multiplying a vector by a scalar 2- Multiplying a vector by a vector a- Scalar product b- Vector product 3- Physical meaning of vector product 2-7 Coordinates notation 2-8 Properties of Vectors Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University  2-1 Introduction Physics deals with a great many quantities that have both magnitude and direction, and it needs a special mathematical language. To describe those quantities, we need the language of vectors. This language are widely used in engineering and many type of sciences. In fact, physics and engineering, need vectors in special ways to explain their phenomena. Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University  2-2 Vectors and Scalars A particle moving along a straight line can move in only two directions. We can take its motion to be positive ( +ve ) in one of these directions and negative ( -ve ) in the other. For a particle moving in three dimensions, however, a plus sign or minus sign is no longer enough to indicate a direction. Instead, we must use a Vector. A vector has magnitude as well as direction. Vectors follow certain rules of combination. Some physical quantities that having both a magnitude and a direction, and thus can be represented with a vector Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University Such as: displacement, velocity, and acceleration. Not all physical quantities involving a direction, for example: Temperature, energy, and mass. Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University  2-3 Adding Vectors Geometrically Let : 𝐚 and Ԧ𝐛 are two vectors, and 𝐬Ԧ = 𝐚 + Ԧ𝐛 𝐚 Ԧ𝐛 𝐬Ԧ = 𝐚 + Ԧ𝐛 Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University  Properties of Vectors 1- Commutative law 𝐚 + Ԧ𝐛 = Ԧ𝐛 + 𝐚 2- Associative law 𝐚 + Ԧ𝐛 + 𝐜Ԧ = 𝐚 + Ԧ𝐛 + 𝐜Ԧ 3- Let Ԧ𝐛 is a vector, then − Ԧ𝐛 is a vector with the same magnitude as Ԧ𝐛, but in the opposite direction. Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University 4- Vector Subtraction Let : 𝐝Ԧ = 𝐚 − Ԧ𝐛 This means: 𝐝Ԧ = 𝐚 + − Ԧ𝐛 Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University  2-4 Components of Vectors A component of vector is the Projection of the vector on x-axis and y-axis. The projection of the vector on x-axis is: x-component The projection of the vector on y-axis is: y-component Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University Then: 𝐚𝐱 = 𝐚 𝐜𝐨𝐬 𝛉 and 𝐚𝐲 = 𝐚 𝐬𝐢𝐧 𝛉 Then: 𝐚𝟐𝐱 + 𝐚𝟐𝐲 = 𝐚𝟐 𝐜𝐨𝐬 𝟐 𝛉 + 𝐚𝟐 𝐬𝐢𝐧𝟐 𝛉 𝐚𝟐𝐱 + 𝐚𝟐𝐲 = 𝐚𝟐 𝐜𝐨𝐬 𝟐 𝛉 + 𝐬𝐢𝐧𝟐 𝛉 = 𝐚𝟐 𝐚𝟐 = 𝐚𝟐𝐱 + 𝐚𝟐𝐲 𝐚= 𝐚𝟐𝐱 + 𝐚𝟐𝐲 Where: 𝐚 is the magnitude of the vector 𝐚: 𝐚 = 𝐚 = 𝐚𝟐𝐱 + 𝐚𝟐𝐲 𝐚𝐲 And the direction of vector 𝐚 is given by: 𝛉 = 𝐭𝐚𝐧−𝟏 𝐚𝐱 𝛉 : is the magnitude of the angle ( direction ) of the vector from the positive x-axis Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University  2-5 Unit Vector A unit vector is a vector that has a magnitude of exactly 1 and points in a particular direction. We can express a vector 𝐚 in terms of a unit vector as: 𝐚 = 𝐚𝐱 𝐢Ƹ + 𝐚𝐲 𝐣Ƹ 𝐢Ƹ is a unit vector along x-axis. 𝐣Ƹ is a unit vector along y-axis. 𝐚𝐱 : is a scalar component of vector 𝐚 along x-axis 𝐚𝐲 : is a scalar component of vector 𝐚 along y-axis 𝐚𝐱 𝐢Ƹ : is a vector component of vector 𝐚 along x-axis 𝐚𝐲 𝐣Ƹ : is a vector component of vector 𝐚 along y-axis 𝐢Ƹ = 𝟏 and 𝐣Ƹ = 𝟏 The Unit Vectors along axes Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University If vector 𝐚 is in 3.Dimension, then we can express 𝐚 as: መ 𝐚 = 𝐚𝐱 𝐢Ƹ + 𝐚𝐲 𝐣Ƹ + 𝐚𝒛 𝐤 Where: 𝐚𝐱 , 𝐚𝐲 , 𝐚𝐧𝐝 𝐚𝐳 : are scalars 𝐢Ƹ is a unit vector along x-axis 𝐣Ƹ is a unit vector along y-axis መ is a unit vector along z-axis 𝐤 𝐚𝐱 : is a scalar component of vector 𝐚 along x-axis The Unit Vectors along axes 𝐚𝐲 : is a scalar component of vector 𝐚 along y-axis 𝐚𝐳 : is a scalar component of vector 𝐚 along z-axis 𝐚𝐱 𝐢Ƹ : is a vector component of vector 𝐚 along x-axis 𝐚𝐲 𝐣:Ƹ is a vector component of vector 𝐚 along y-axis መ is a vector component of vector 𝐚 along z-axis 𝐚𝒛 𝐤: መ = 𝟏 𝐢Ƹ = 𝐣Ƹ = 𝐤 Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University Example: A small airplane leaves an airport on an overcast day and is later sighted 215 km away, in a direction making an angle of 22° east of due north. How far east and north is the airplane from the airport when sighted? Answer: 𝐝Ԧ = 𝐝𝐱 𝐢Ƹ + 𝐝𝐲 𝐣Ƹ dx = d cos (ϑ ) = (215 km). cos (68°) = 81 km dy = d sin (ϑ ) = (215 km). sin (68°) = 200 km Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University Example: From below figure, find the resultant vector, where: the magnitude of each d vector equal 6 unit. Answer: 𝐝Ԧ𝐫𝐞𝐬 = 𝐝Ԧ𝟏 + 𝐝Ԧ𝟐 + 𝐝Ԧ𝟑 + 𝐝Ԧ𝟒 + 𝐝Ԧ𝟓 To add these vectors, we must find their net x-components and the net y- components. Then: Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University  𝐝Ԧ𝐫𝐞𝐬 = 𝐝Ԧ = 𝐢Ƹ 𝐝𝐫𝐞𝐬 ȁ𝐱 + 𝐣Ƹ 𝐝𝐫𝐞𝐬 ȁ𝐲 In general: 𝐝Ԧ = 𝐢Ƹ 𝐝𝐱 + 𝐣Ƹ 𝐝𝐲 𝐝𝐱 = 𝐝 𝐜𝐨𝐬 𝛉 and 𝐝𝐲 = 𝐝 𝐬𝐢𝐧 𝛉 𝐝𝐲 d= 𝐝Ԧ = 𝐝𝟐𝐱 + 𝐝𝟐𝐲 and 𝛉 = 𝐭𝐚𝐧−𝟏 𝐝𝐱 𝐝𝐫𝐞𝐬 ȁ𝐱 = 𝐝𝟏𝐱 + 𝐝𝟐𝐱 + 𝐝𝟑𝐱 + 𝐝𝟒𝐱 + 𝐝𝟓𝐱 𝐝𝐫𝐞𝐬 ȁ𝐲 = 𝐝𝟏𝐲 + 𝐝𝟐𝐲 + 𝐝𝟑𝐲 + 𝐝𝟒𝐲 + 𝐝𝟓𝐲 Magnitude of each d vector = 6 unit Then: Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University  𝐝𝟏𝐱 = 𝐝𝟏 𝐜𝐨𝐬 𝛉𝟏 = 𝟔 𝐜𝐨𝐬 𝟎𝐎 = + 𝟔 𝐜𝐦 𝐝𝟐𝐱 = 𝐝𝟐 𝐜𝐨𝐬 𝛉𝟐 = 𝟔 𝐜𝐨𝐬 (𝟏𝟓𝟎𝐎 ) = −𝟓. 𝟐 𝐜𝐦 𝐝𝟑𝐱 = 𝐝𝟑 𝐜𝐨𝐬 𝛉𝟑 = 𝟔 𝐜𝐨𝐬 (𝟏𝟖𝟎𝐎 ) = − 𝟔 𝐜𝐦 𝐝𝟒𝐱 = 𝐝𝟒 𝐜𝐨𝐬 𝛉𝟒 = 𝟔 𝐜𝐨𝐬 (𝟐𝟒𝟎𝐎 ) = −𝟑 𝐜𝐦 𝐝𝟓𝐱 = 𝐝𝟓 𝐜𝐨𝐬 𝛉𝟓 = 𝟔 𝐜𝐨𝐬 (𝟗𝟎𝐎 ) = 𝟎 𝐜𝐦 Note that: 𝐬𝐢𝐧 𝟎 = 𝟎 𝐬𝐢𝐧 𝟗𝟎 = 𝟏 𝐜𝐨𝐬 𝟎 = 𝟏 𝐜𝐨𝐬 𝟗𝟎 = 𝟎 Then: 𝐝𝐫𝐞𝐬 ȁ𝐱 = +𝟔 + −𝟓. 𝟐 + −𝟔 + −𝟑 + 𝟎 = −𝟖. 𝟐 𝐜𝐦 -ve sign means that: the component is in the –ve x-direction, Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University Now: 𝐝𝟏𝐲 = 𝐝𝟏 𝐬𝐢𝐧 𝛉𝟏 = 𝟔 𝐬𝐢𝐧 (𝟎𝐎 ) = 𝟎 𝐜𝐦 𝐝𝟐𝐲 = 𝐝𝟐 𝐬𝐢𝐧 𝛉𝟐 = 𝟔 𝐬𝐢𝐧 (𝟏𝟓𝟎𝐎 ) = + 𝟑 𝐜𝐦 𝐝𝟑𝐲 = 𝐝𝟑 𝐬𝐢𝐧 𝛉𝟑 = 𝟔 𝐬𝐢𝐧 (𝟏𝟖𝟎𝐎 ) = 𝟎 𝐜𝐦 𝐝𝟒𝐲 = 𝐝𝟒 𝐬𝐢𝐧 𝛉𝟒 = 𝟔 𝐬𝐢𝐧 (𝟐𝟒𝟎𝐎 ) = −𝟓. 𝟐 𝐜𝐦 𝐝𝟓𝐲 = 𝐝𝟓 𝐬𝐢𝐧 𝛉𝟓 = 𝟔 𝐬𝐢𝐧 (𝟗𝟎𝐎 ) = + 𝟔 𝐜𝐦 Then: 𝐝𝐫𝐞𝐬 ȁ𝐲 = 𝟎 + +𝟑 + 𝟎 + −𝟓. 𝟐 + +𝟔 = +𝟑. 𝟖 𝐜𝐦 Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University Then: 𝟐 𝟐 𝐝𝐫𝐞𝐬 = 𝐝𝐫𝐞𝐬 ȁ𝐱 + 𝐝𝐫𝐞𝐬 ȁ𝐲 𝐝𝐫𝐞𝐬 = −𝟖. 𝟐 𝟐 + +𝟑. 𝟖 𝟐 = 𝟗 𝐜𝐦 𝐝𝐫𝐞𝐬 ȁ𝐲 + 𝟑.𝟖 𝛉 = 𝐭𝐚𝐧−𝟏 = 𝐭𝐚𝐧−𝟏 = −𝟐𝟒. 𝟖𝟔 ≅ − 𝟐𝟓𝐨 𝐝𝐫𝐞𝐬 ȁ𝐱 − 𝟖.𝟐 This is mean that the result: is a vectors of length 9 cm and angle -25 degree, i.e. +335 degree , i.e. in the 4th quadrant of the xy-plane. Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University However, when we construct the vector from its components, we see that the direction of 𝐝Ԧ𝐫𝐞𝐬 is in the second quadrant. Thus, we must “fix” the calculator’s answer by adding 180°. i.e. 𝛉 = − 𝟐𝟓𝐨 + 𝟏𝟖𝟎𝐨 ≅ 𝟏𝟓𝟓𝐨 Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University Thus, the 𝐝Ԧ𝐫𝐞𝐬 is given by: 𝐝Ԧ𝐫𝐞𝐬 = 𝟗 𝐜𝐦 𝐚𝐭 𝟏𝟓𝟓𝐨 The direction of 𝐝Ԧ𝐫𝐞𝐬 is 155° from +ve x-axis and of length equal 9 cm from point Home to point Final Vector 𝐝Ԧ𝐡𝐨𝐦 has the same magnitude as: 𝐝Ԧ𝐫𝐞𝐬 but in the opposite direction. Thus, has magnitude and angle: 𝐝Ԧ𝐫𝐞𝐬 = 𝟗 𝐜𝐦 𝐚𝐭 − 𝟐𝟓𝐨 Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University H.W.: In the figure: a- what are the signs of the x-components of vector 𝐝𝟏 and vector 𝐝𝟐 ? b- what are the signs of the y-components of vector 𝐝𝟏 and vector 𝐝𝟐 ? c- what are the sign of the x-component of vector 𝐝𝟏 + 𝐝𝟐 ? d- what are the sign of the y-component of vector 𝐝𝟏 + 𝐝𝟐 ? Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University H.W.: Figure shows the following three vectors: 𝐚 = 𝟒. 𝟐 𝐦 𝐢Ƹ - (1.5 m) 𝐣Ƹ Ԧ𝐛 = −𝟏. 𝟔 𝐦 𝐢Ƹ + (2.9 m) 𝐣Ƹ 𝐜Ԧ = −𝟑. 𝟕 𝐦 ෠𝐣 What is their vector sum 𝐫Ԧ , as shown ? Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University  2-6 Multiplying Vectors There are 3 ways in which vectors can be multiplied, these are: 1- Multiplying a vector by a scalar 2- Multiplying a vector by a vector, has 2 types a- Scalar Product b- Vector Product Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University  1- Multiplying a vector by a scalar Let: 𝐝Ԧ is a vector, 𝛂 is a scalar , then: 𝐑 = 𝛂 𝐝Ԧ = 𝛂 𝐢Ƹ 𝐝𝐱 + 𝐣Ƹ 𝐝𝐲 = 𝐢Ƹ 𝛂 𝐝𝐱 + 𝐣Ƹ 𝛂 𝐝𝐲 𝐑 = 𝐢Ƹ 𝐫𝐱 + 𝐣Ƹ 𝐫𝐲 Where: 𝐑 is in the same direction as 𝐝Ԧ , but with magnitude multiplied by value 𝛂 𝐫𝐱 = 𝛂 𝐝𝐱 𝐫𝐱 is in the same direction of 𝐝𝐱 𝐫𝐲 = 𝛂 𝐝𝐲 𝐫𝐲 is in the same direction of 𝐝𝐲 Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University  2- Scalar product The Scalar product or dot product between 2 vectors: is an algebraic operation that takes two vectors and return a single number. The physical meaning of dot product has the geometric interpretation as: the length of the projection onto the unit vector when the two vectors are placed so that their tails coincide. Scalar product (Dot product) can be used for: computing the angle θ between two multiplied vectors The mathematical representation of scalar product is given by: 𝐚. Ԧ𝐛 = 𝐒𝐜𝐚𝐥𝐚𝐫 𝐪𝐮𝐚𝐧𝐭𝐢𝐭𝐲 ( 𝐑𝐞𝐚𝐝 𝐚𝐬: 𝐚 𝐝𝐨𝐭 𝐛 ) Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University Where: 𝐚. Ԧ𝐛 = 𝐚 Ԧ𝐛 𝐜𝐨𝐬 (𝛉) 𝐚 = 𝐚 = 𝐦𝐚𝐠𝐧𝐢𝐭𝐮𝐝𝐞 𝐨𝐟 𝐚 𝐛 = Ԧ𝐛 = 𝐦𝐚𝐠𝐧𝐢𝐭𝐮𝐝𝐞 𝐨𝐟 Ԧ𝐛 𝛉 : is the angle between the direction of vector 𝐚 and direction of vector 𝐛 Note: There are actually two angles verify the equation of dot product, these angles are: 𝛉 and (𝟑𝟔𝟎 − 𝛉) , because their cosines are the same. Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University A dot product can be regarded as the product of two quantities: (1) the magnitude of one of the vectors and (2) the scalar component of the second vector along the direction of the first vector. i.e.: 𝐚. Ԧ𝐛 = 𝐚 Ԧ𝐛 𝐜𝐨𝐬 𝛉 = 𝐚. Ԧ𝐛 𝐜𝐨𝐬 (𝛉) = Ԧ𝐛. 𝐚 𝐜𝐨𝐬 𝛉 For example in below figure, 𝐚 has a scalar component ( 𝐚 𝐜𝐨𝐬 𝛉) along the direction of Ԧ𝐛 , Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University Similarly, Ԧ𝐛 has a scalar component ( 𝐛 𝐜𝐨𝐬 𝛉) along the direction of 𝐚. i.e., 𝐚. Ԧ𝐛 = 𝐚 Ԧ𝐛 𝐜𝐨𝐬 (𝛉) = 𝐚 𝐛 𝐜𝐨𝐬 (𝛉) = 𝐚 𝐜𝐨𝐬 𝛉 𝐛 = 𝐚 𝐛 𝐜𝐨𝐬 𝛉 Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University Properties of Scalar Product First Property: Commutative property 𝐚. Ԧ𝐛 = Ԧ𝐛. 𝐚 i.e. the Scalar product is commutative property Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University Second Property: (A) if the angle 𝛉 = 𝟎 , then: 𝐜𝐨𝐬 𝟎 = 𝟏 𝐚. Ԧ𝐛 = 𝐚 Ԧ𝐛 𝐜𝐨𝐬 (𝛉) = 𝐚 𝐛 𝐜𝐨𝐬 (𝟎) = 𝐚 𝐛 = 𝐦𝐚𝐱𝐢𝐦𝐮𝐦 i.e. the component of one vector along the other is maximum, i.e. the dot product is maximum (B) If the angle 𝛉 = 90°, then: 𝐜𝐨𝐬 𝟗𝟎 = 𝟎 𝐚. Ԧ𝐛 = 𝐚 Ԧ𝐛 𝐜𝐨𝐬 (𝛉) = 𝐚 𝐛 𝐜𝐨𝐬 (𝟗𝟎) = 𝐳𝐞𝐫𝐨 i.e. the component of one vector along the other is zero, i.e. the dot product is minimum (C) If the angle 𝛉 = 180°, then: 𝐜𝐨𝐬 𝟏𝟖𝟎 = −𝟏 𝐚. Ԧ𝐛 = 𝐚 Ԧ𝐛 𝐜𝐨𝐬 𝛉 = 𝐚 𝐛 𝐜𝐨𝐬 𝟏𝟖𝟎 = − 𝐚𝐛 What does this mean ? Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University Third: 𝐚. Ԧ𝐛 = 𝐢Ƹ 𝐚𝐱 + 𝐣Ƹ 𝐚𝐲 + 𝐤 መ 𝐚𝐳. 𝐢Ƹ 𝐛𝐱 + 𝐣Ƹ 𝐛𝐲 + 𝐤 መ 𝐛𝐳 𝐚. Ԧ𝐛 = 𝐢Ƹ 𝐚𝐱. 𝐢Ƹ 𝐛𝐱 + 𝐣Ƹ 𝐛𝐲 + 𝐤 መ 𝐛𝐳 + 𝐣Ƹ 𝐚𝐲. 𝐢Ƹ 𝐛𝐱 + 𝐣Ƹ 𝐛𝐲 + 𝐤 መ 𝐛𝐳 + 𝐤 መ 𝐚𝐳. 𝐢Ƹ 𝐛𝐱 + 𝐣Ƹ 𝐛𝐲 + 𝐤 መ 𝐛𝐳 𝐚. Ԧ𝐛 መ 𝐚𝐱 𝐛𝐳 + 𝐣.Ƹ 𝐢Ƹ 𝐚𝐲 𝐛𝐱 + 𝐣.Ƹ 𝐣Ƹ 𝐚𝐲 𝐛𝐲 + 𝐣.Ƹ 𝐤 = 𝐢.Ƹ 𝐢 𝐚𝐱 𝐛𝐱 + 𝐢.Ƹ 𝐣Ƹ 𝐚𝐱 𝐛𝐲 + 𝐢.Ƹ 𝐤 መ 𝐚 𝐲 𝐛𝐳 መ 𝐢Ƹ 𝐚𝐳 𝐛𝐱 + 𝐤. + 𝐤. መ 𝐣Ƹ 𝐚𝐳 𝐛𝐲 + 𝐤. መ 𝐤መ 𝐚𝐳 𝐛𝐳 𝐚. Ԧ𝐛 = 𝐚𝐱 𝐛𝐱 + 𝐚𝐲 𝐛𝐲 + 𝐚𝐳 𝐛𝐳 where: መ =𝟏 𝐢Ƹ = 𝐣Ƹ = 𝐤 𝐢.Ƹ 𝐢Ƹ = 𝐣.Ƹ 𝐣Ƹ = 𝐤. መ 𝐤መ = 1 𝐜𝐨𝐬 𝟎 = 𝟏 𝐜𝐨𝐬 𝟗𝟎 = 𝟎 𝐢.Ƹ 𝐣Ƹ = 𝐣.Ƹ 𝐢Ƹ = 𝐢.Ƹ 𝐤 መ = 𝐤. መ 𝐢Ƹ = 𝐣.Ƹ 𝐤 መ = 𝐤. መ 𝐣Ƹ = 𝐳𝐞𝐫𝐨 𝐢Ƹ. 𝐢Ƹ = 𝐢Ƹ 𝐢Ƹ 𝐜𝐨𝐬 𝟎 = 𝟏 𝐣.Ƹ 𝐣Ƹ = 𝐣Ƹ 𝐣Ƹ 𝐜𝐨𝐬 𝟎 = 𝟏 መ 𝐤 𝐤. መ = 𝐤 መ 𝐤 መ 𝐜𝐨𝐬 𝟎 = 𝟏 𝐢Ƹ. 𝐣Ƹ = 𝐢Ƹ 𝐣Ƹ 𝐜𝐨𝐬 𝟗𝟎 = 𝟎 Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University Example: What is the angle 𝛉 between vector 𝐚 and vector Ԧ𝐛: 𝐚 = 𝟑𝐢Ƹ − 𝟒𝐣Ƹ , Ԧ𝐛 = −𝟐𝐢Ƹ + 𝟑 𝐤 መ Answer: 𝐚. Ԧ𝐛 = 𝐚 Ԧ𝐛 𝐜𝐨𝐬 𝛉 = 𝐚 𝐛 𝐜𝐨𝐬 𝛉 𝐚= 𝐚 = 𝐚𝟐𝐱 + 𝐚𝟐𝐲 + 𝐚𝟐𝐳 = 𝟑 𝟐 + (− 4)2 + 𝟎 = 𝟓 𝐛 = Ԧ𝐛 = 𝐛𝐱𝟐 + 𝐛𝐲𝟐 + 𝐛𝐳𝟐 = −𝟐 𝟐 + 𝟎 + (+3)2 = 𝟑. 𝟔𝟏 𝐚. Ԧ𝐛 = 𝐚𝐱 𝐛𝐱 + 𝐚𝐲 𝐛𝐲 + 𝐚𝐳 𝐛𝐳 = 𝟑 𝐱 − 𝟐 + −𝟒 𝐱 𝟎 + 𝟎 𝐱 𝟑 = −𝟔 −𝟔 -6 = ( 5 ). ( 3.61 ). 𝐜𝐨𝐬 (𝛉) then: 𝛉 = 𝐜𝐨𝐬 −𝟏 𝟓 𝐱 𝟑.𝟔𝟏 𝛉 = 𝟏𝟏𝟎 𝐎 Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University H.W: Prove that the dot product has commutative property, i.e.: 𝐚. Ԧ𝐛 = Ԧ𝐛. 𝐚 H.W: Vectors 𝐂Ԧ and 𝐃 have magnitudes 3 units and 4 units, respectively. What is the angle between the direction 𝐂Ԧ and 𝐃 , if 𝐂Ԧ. 𝐃 equal to: 1. zero 2. ( 12 ) units 3. ( -12 ) units Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University  3- Vector product The vector product of 𝐚 and Ԧ𝐛 , written 𝐜Ԧ = 𝐚 𝐱 Ԧ𝐛 = 𝐕𝐞𝐜𝐭𝐨𝐫 𝐪𝐮𝐚𝐧𝐭𝐢𝐭𝐲 ( 𝐑𝐞𝐚𝐝: 𝐚 𝐜𝐫𝐨𝐬𝐬 𝐛 ) produces a third vector 𝐜Ԧ whose magnitude is: 𝐜Ԧ = 𝐚 𝐱 Ԧ𝐛 = 𝐚 Ԧ𝐛 𝐬𝐢𝐧 𝛗 = 𝐚𝐛 𝐬𝐢𝐧 𝛗 Where: 𝛗 is the smaller angle between 𝐚 and Ԧ𝐛. You must use the smaller of the two angles between the vectors, because 𝐬𝐢𝐧 𝛗 and 𝐬𝐢𝐧 (𝟑𝟔𝟎 − 𝛗 ) differ in algebraic sign. Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University  4- Physical meaning of vector product The vector product of 𝐚 and Ԧ𝐛, written 𝐜Ԧ = 𝐚 𝐱 Ԧ𝐛 𝐜Ԧ is a vector perpendicular to both 𝐚 and Ԧ𝐛, and thus: normal to the plane containing the vector 𝐚 and vector Ԧ𝐛. Vector product has many applications in mathematics, physics, engineering, and computer programming. Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University More generally, the magnitude of the cross product equals: the area of a parallelogram with the vectors four sides. in particular, the magnitude of the cross product of two perpendicular vectors equal the product of their lengths. Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University  2-7 Coordinates Notation The standard basis vectors i, j, and k 𝐢Ƹ 𝐱 𝐣Ƹ = 𝐤መ መ = 𝐢Ƹ 𝐣Ƹ 𝐱 𝐤 መ 𝐱 𝐢Ƹ = 𝐣Ƹ 𝐤 𝐣Ƹ 𝐱 𝐢Ƹ = − 𝐤 መ መ 𝐱 𝐣Ƹ = − 𝐢Ƹ 𝐤 መ = − 𝐣Ƹ 𝐢Ƹ 𝐱 𝐤 መ 𝐱𝐤 𝐢Ƹ 𝐱 𝐢Ƹ = 𝐣Ƹ 𝐱 𝐣Ƹ = 𝐤 መ =𝟎 𝐢. 𝐞. 𝐳𝐞𝐫𝐨 𝐯𝐞𝐜𝐭𝐨𝐫 𝐚 𝐱 Ԧ𝐛 = 𝐢Ƹ 𝐚𝐱 + 𝐣Ƹ 𝐚𝐲 + 𝐤 መ 𝐚𝐳 𝐱 𝐢Ƹ 𝐛𝐱 + 𝐣Ƹ 𝐛𝐲 + 𝐤 መ 𝐛𝐳 𝐚 𝐱 Ԧ𝐛 መ + 𝐚𝐲 𝐛𝐱 𝐣Ƹ 𝐱 𝐢Ƹ + 𝐚𝐲 𝐛𝐲 𝐣Ƹ 𝐱 𝐣Ƹ + 𝐚𝐲 𝐛𝐳 𝐣Ƹ 𝐱 𝐤 = 𝐚𝐱 𝐛𝐱 𝐢Ƹ 𝐱 𝐢Ƹ + 𝐚𝐱 𝐛𝐲 𝐢Ƹ 𝐱 𝐣Ƹ + 𝐚𝐱 𝐛𝐳 𝐢Ƹ 𝐱 𝐤 መ + 𝐚𝐳 𝐛𝐱 𝐤መ 𝐱 𝐢Ƹ + 𝐚𝐳 𝐛𝐲 𝐤 መ 𝐱 𝐣Ƹ + 𝐚𝐳 𝐛𝐳 𝐤መ 𝐱𝐤መ 𝐚 𝐱 Ԧ𝐛 መ + 𝐚𝐲 𝐛𝐱 𝐣Ƹ 𝐱 𝐢Ƹ + 𝐚𝐲 𝐛𝐳 𝐣Ƹ 𝐱 𝐤 = 𝐚𝐱 𝐛𝐲 𝐢Ƹ 𝐱 𝐣Ƹ + 𝐚𝐱 𝐛𝐳 𝐢Ƹ 𝐱 𝐤 መ + 𝐚𝐳 𝐛𝐱 𝐤 መ 𝐱 𝐢Ƹ + 𝐚𝐳 𝐛𝐲 𝐤 መ 𝐱 𝐣Ƹ 𝐚 𝐱 Ԧ𝐛 = 𝐚𝐱 𝐛𝐲 𝐤 መ + 𝐚𝐱 𝐛𝐳 −𝐣Ƹ + 𝐚𝐲 𝐛𝐱 −𝐤 መ + 𝐚𝐲 𝐛𝐳 +𝐢Ƹ + 𝐚𝐳 𝐛𝐱 𝐣Ƹ + 𝐚𝐳 𝐛𝐲 −𝐢Ƹ 𝐚 𝐱 Ԧ𝐛 = 𝐢Ƹ ( 𝐚𝐲 𝐛𝐳 − 𝐚𝐳 𝐛𝐲 ) − 𝐣Ƹ ( 𝐚𝐱 𝐛𝐳 − 𝐚𝐳 𝐛𝐱 ) + 𝐤 መ ( 𝐚𝐱 𝐛𝐲 − 𝐚𝐲 𝐛𝐱 ) Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University Unit vectors 𝐢, 𝐣, 𝐚𝐧𝐝 𝐤 and vector components of vector 𝐚: 𝐚𝐱 , 𝐚𝐲 , 𝐚𝐧𝐝 𝐚𝐳 Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University  Properties of Cross Product Property-1: cross product is not commute 𝐜Ԧ = 𝐚 𝐱 Ԧ𝐛 = − Ԧ𝐛 𝐱 𝐚 Property-2: If 𝐚 and Ԧ𝐛 are parallel or antiparallel, i.e. 𝛗 = 𝐳𝐞𝐫𝐨 or 𝛗 = 𝟏𝟖𝟎 , and: 𝐬𝐢𝐧 𝛗 = 𝟎 , then: 𝐚 𝐱 Ԧ𝐛 = 𝐳𝐞𝐫𝐨 i.e. If two vectors have the same direction (or have the exact opposite direction from one another), i.e. are not linearly independent, Then the cross product is zero Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University Property-3: If 𝐚 and Ԧ𝐛 are perpendicular to each other, i.e. 𝛗 = 𝟗𝟎 , then: 𝐜Ԧ = 𝐚 𝐱 Ԧ𝐛 = 𝐦𝐚𝐱𝐢𝐦𝐮m i.e. the magnitude of 𝐚 𝐱 Ԧ𝐛, which can be written as: 𝐚 𝐱 Ԧ𝐛 , is maximum when 𝐚 and Ԧ𝐛 are perpendicular to each other. i.e. The direction of 𝐜Ԧ is perpendicular to the plane contains: 𝐚 and Ԧ𝐛 Property-4: Cross product in unit vector is given by: 𝐚 𝐱 Ԧ𝐛 = 𝐢Ƹ 𝐚𝐱 + 𝐣Ƹ 𝐚𝐲 + 𝐤 መ 𝐚𝐳 𝐱 𝐢Ƹ 𝐛𝐱 + 𝐣Ƹ 𝐛𝐲 + 𝐤 መ 𝐛𝐳 Property-5 መ 𝐱𝐤 𝐢Ƹ 𝐱 𝐢Ƹ = 𝐣Ƹ 𝐱 𝐣Ƹ = 𝐤 መ = 𝐳𝐞𝐫𝐨 Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University Property 6: መ 𝐢Ƹ 𝐱 𝐣Ƹ = 𝐤 መ = 𝐢Ƹ 𝐣Ƹ 𝐱 𝐤 መ 𝐱 𝐢Ƹ = 𝐣Ƹ 𝐤 መ 𝐣Ƹ 𝐱 𝐢Ƹ = − 𝐤 መ 𝐱 𝐣Ƹ = − 𝐢Ƹ 𝐤 መ = − 𝐣Ƹ 𝐢Ƹ 𝐱 𝐤 Property 7: 𝐢 𝐣 𝐤 መ = 𝐚ො 𝐱 𝐛 𝐚𝐱 𝐚𝐲 𝐚𝐳 = 𝐛𝐱 𝐛𝐲 𝐛𝐳 𝐚 𝐱 Ԧ𝐛 = 𝐢Ƹ 𝐚𝐲 𝐛𝐳 − 𝐚𝐳 𝐛𝐲 መ 𝐚𝐱 𝐛𝐲 − 𝐚𝐲 𝐛𝐱 − 𝐣Ƹ 𝐚𝐱 𝐚𝐳 − 𝐚𝐳 𝐚𝐱 + 𝐤 Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University H.W.: Two vectors 𝐚 and vector Ԧ𝐛: 𝐚 = 𝟑𝐢Ƹ − 𝟒𝐣Ƹ , መ what is : 𝐜Ԧ Ԧ𝐛 = −𝟐𝐢Ƹ + 𝟑 𝐤, = 𝐚 𝐱 Ԧ𝐛 Answer: መ 𝐜Ԧ = −𝟏𝟐 𝐢Ƹ − 𝟗 𝐣Ƹ − 𝟖 𝐤 Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University H.W.: Vector 𝐂Ԧ has magnitude of 3 unit and vector 𝐃 has magnitudes of unit 4. What is the angle between the directions 𝐂Ԧ and 𝐃 if the magnitude of the vector product 𝐂Ԧ 𝐱 𝐃 is: a- zero b- 12 units Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University Question: Vector 𝐚 lies in the xy-plane, has a magnitude of 18 units and points in a direction 250° from the +ve direction of the x-axis. Also, vector Ԧ𝐛 has a magnitude of b 12 units and points in the +ve direction of the z-axis. What is the vector product: 𝐜Ԧ = 𝐚 𝐱 Ԧ𝐛. Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University H.W.: Page 53-54 Q-1 Q-2 Q-4 Q-6 Q-7 Q-8 Q-9 Q-10 H.W.: SECT. 3-4, Page 54 *1 *2 *3 *4 *5 H.W.: SECT. 3-6, Page 54 *9 *10 *11 Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University  The End of Chapter 1 Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University  Report including Title 1. Report present by each student. Presented by: 2. Each report includes 15-20 pages, using Department, College, University Standard Form: Contents  English Language Ch-1 Introduction  Font: Times New Roman Ch-2 Theory of the Subject  Font Size: 14 Ch-3 Applications  Spacing: 1.5 Ch-4 Discussion  References: Not less than 3 References Ch-5 Conclusions 3. Subject of the Report Only in the Field of References Physic. Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University  Titles of Reports 1. General Relativity 2. Special Relativity 3. Physics of Electron 4. Physics of Proton 5. Physics of Neutron 6. Plasma Physics 7. Physics of the Atmosphere 8. Physics of the Ionosphere 9. Physics of LASER 10. Physics of MASER 11. Physics of RADAR 12. Physics of LADAR 13. Quantum Physics 14. Microscope 15. Electron Microscope 16. Optical Telescopes 17. Millemetric Telescope 18. Wind Energy 18. Solar Energy 19. Physics of Doppler’s Effect 20. Theories of Atomic Structure Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University 18. Theories of Nuclear Structure 19. Theories of Gravity 20. Sound waves 21. Theories of Light’s Nature 22. Application of Electro-Magnetic Waves 23. Nuclear Radiation 24. Maxwell’s Equations 25. Satellites Communication 26. Information Theory 27. Communication Theory 28. Molecular Physics 29. Fiber Communication 30. Microwave Communication 31. Origin of the Universe 32. Quantum Entanglement 33. Quantum Superposition 34. Evolution of the Light Theories Prof. Dr. Ayad A. Al-Ani, Professor of Digital Image Processing College of Information Engineering (COIE), Al-Nahrain University