Chapter 3: VECTORS 3-2 Vectors and Scalars 3-2 Vectors and Scalars Is a number with units. It can be positive or negative. Example: distance, mass, speed, Temperature… Vector Is a quantity with both direction and magnitude. Example: velocity, force, displacement…

3-2 Vectors and Scalars The figure shows a displacement vector where B the particle undergoes a displacement from A to B. A B’ The displacement vector is represented by B” A’ an arrow pointing from A to B. (we will use A” here triangle arrowheads for arrows that represent vectors) In the figure, the arrows from A to B, from A’ to B’, and from A” to B” have the same magnitude and direction. They represent identical displacement vectors.

3-2 Vectors and Scalars B A The displacement vector tells us nothing about the actual path that the particle takes. In the figure, for example, all three paths connecting points A and B correspond to the same displacement vector. B A

3-3 Adding Vectors Geometrically Vector is written with an arrow over the symbol 𝒂 . The magnitude of the vector 𝒂 is written as 𝒂 . ff 𝒂 𝒃 tail head 𝒂 𝒃 𝒔 = 𝒂 + 𝒃

3-3 Adding Vectors Geometrically Two important properties: Adding to gives the same result as adding to , that is, (commutative law) Start Finish

3-3 Adding Vectors Geometrically 2. For more than two vectors, the addition can be done in any order. That is: (associative law)

3-3 Adding Vectors Geometrically The vector − 𝒂 is a vector with the same magnitude as 𝒂 but the opposite direction. − 𝒂 𝒂 Vector substraction : 𝒅 = 𝒂 − 𝒃 = 𝒂 + − 𝒃 𝒂 𝒃 𝒂 − 𝒃 𝒅 = 𝒂 − 𝒃

3-4 Components of vectors Set up two dimensional coordinate system. 𝒙 𝒚 𝒐 𝒂 𝒂 𝒙 𝒂 𝒚 (𝒙 Component) (𝒚 Component) 𝜽 is measured relative to the 𝒙 axis.

form a the right triangle shown in figure. The vector is completely determined by either: the pair ( ) or the pair ( ). It is possible to obtain any of these pairs in terms of the other pair using the right triangle as follows:

3-4 Components of vectors Given the magnitude 𝒂 and direction 𝜽 of a vector 𝒂 , we get its component 𝒂 𝒙 and 𝒂 𝒚 . 𝒂 𝒙 =𝒂 𝒄𝒐𝒔 𝜽 𝒂 𝒚 =𝒂 𝒔𝒊𝒏 𝜽 Given the components 𝒂 𝒙 and 𝒂 𝒚 of a vector 𝒂 , we get its magnitude 𝒂 and direction 𝜽. 𝒂= 𝒂 𝒙 𝟐 + 𝒂 𝒚 𝟐 𝜽= 𝐭𝐚𝐧 −𝟏 𝒂 𝒚 𝒂 𝒙

3-5 Unit vectors A unit vector is a vector that has a magnitude of exactly 1 and points in a particular direction. The unit vectors in the positive directions of the x, y and z directions are labeled , and , respectively (see figure)

are called the vector components of . 3-5 Unit vectors In two dimensions any vector like can be written as: 𝒙 Component 𝒚 Component are called the vector components of . For example, if a vector 𝒂 has the scalar component 𝒂 𝒙 =𝟓 𝒎 and 𝒂 𝒚 =𝟑 𝒎, we write it as: 𝒂 = 𝟓 𝒎 𝒊 + 𝟑 𝒎 𝒋

3-6 Adding vectors by components If a vector 𝒂 is written as: 𝒂 = 𝒂 𝒙 𝒊 + 𝒂 𝒚 𝒋 and 𝒃 = 𝒃 𝒙 𝒊 + 𝒃 𝒚 𝒋 Then 𝒓 = 𝒂 + 𝒃 = 𝒂 𝒙 + 𝒃 𝒙 𝒊 + 𝒂 𝒚 + 𝒃 𝒚 𝒋 The component of 𝒓 : 𝒓 𝒙 =𝒂 𝒙 + 𝒃 𝒙 𝒓 𝒚 =𝒂 𝒚 + 𝒃 𝒚

To Add vectors and by using components method 1. Calculate the scalar components: , , and . Calculate the components of the sum : Combine the components of to get itself: For vector subtraction: and

( the magnitude of )x(absolute value of ) 3-8 Multiplying Vectors Multiplying a Vector by a Scalar: Multiplication of a vector 𝒂 by a scalar 𝐬 gives a vector 𝐬 𝒂 . If 𝑠>0 : 𝒂 and 𝐬 𝒂 have the same direction. If 𝑠<0 : 𝒂 and 𝐬 𝒂 have the opposite direction. The magnitude of = ( the magnitude of )x(absolute value of )

3-8 Multiplying Vectors 𝒂 . 𝒃 =𝒂 𝒃 𝒄𝒐𝒔 ∅ The Scalar or Dot Product: The scalar product of the vectors 𝒂 and 𝒃 is written as : 𝒂 . 𝒃 𝒂: magnitude of 𝒂 𝒃: magnitude of 𝒃 ∅: angle between 𝒂 and 𝒃 𝒂 . 𝒃 =𝒂 𝒃 𝒄𝒐𝒔 ∅ 𝒂 𝒃 ∅

= the component of along the direction of = the product of the magnitude of one of the vectors by the scalar component of the second vector along the direction of the first vector (see figure): = the component of along the direction of Notice that: Component of 𝑎 along direction of 𝑏 is acos𝜃 𝒂 𝒃 𝜽 Component of 𝑏 𝑎 is bcos𝜃

In unit vector notation, we write: Each vector component of the first vector is to be dotted by each vector component of the second vector. For example: and Where the angle between and is 0 and that between and is 90. By doing so, we get:

Consider the case of two dimensional vectors and In terms of their components, the dot (scalar) product of two vectors 𝒂 and 𝒃 is written as : 𝒂 = 𝒂 𝒙 𝒊 + 𝒂 𝒚 𝒋 and 𝒃 = 𝒃 𝒙 𝒊 + 𝒃 𝒚 𝒋 𝒂 . 𝒃 = 𝒂 𝒙 𝒃 𝒙 + 𝒂 𝒚 𝒃 𝒚

3-8 Multiplying Vectors . Here is the smaller angle between and The Vector or Cross Product: The vector product or cross product of the vectors 𝒂 and 𝒃 is written as : 𝒂 × 𝒃 and is another vector 𝒄 , where 𝒄 = 𝒂 × 𝒃 . The magnitude of 𝒄 is defined by 𝑪= 𝒂 × 𝒃 =𝒂 𝒃 𝒔𝒊𝒏∅ The vector 𝒄 is perpendicular to the plane of the two vectors 𝒂 and 𝒃 The direction of 𝒄 is given by the right hand rule. . Here is the smaller angle between and

Right Hand Rule for Vector Product: To obtain the direction of vector , where we use the right-hand rule as follows (see figure on page 28): Place the vectors and tail to tail Imagine a line that is perpendicular to their plane where they meet. Place your right hand around that line in such a way that your fingers would sweep into through the smaller angle between them. Your thumb points in the direction of .

Right Hand Rule for Vector Product: By using the right-hand rule we can see that the direction of , where is opposite to the direction of . So, that is: as the figure shows. Example: By using right-hand rule we find that the vector points in the direction, therefore:

Right Hand Rule for Vector Product:

In unit vector notation we write: Here each vector component of the first vector is to be crossed by each vector component of the second vector. Notice that: and so on. By doing so we get: This can be obtained using the determinant, as follows:

3-8 Multiplying Vectors

3-7 Vectors and the Laws of Physics If we rotate the axes (but not the vector ), through an angle as in the figure the components will have new values say, Since there are infinite number of choices of there are an infinite number of different pairs of components of . Each of these pairs produce the same magnitude and direction for the vector. In the figure, we have: We thus have freedom in choosing the coordinate system. This is also true of the relations (laws) of physics; they are all independent of the choice of the coordinate system.