VECTORS Vectors in 2- D and 3- D in Euclidean space or flatland are easy compared to vectors in non- Euclidean space. In Cartesian coordinates we write a component of a vector as where the index i stands for either x, y or z. For instance, represents the y- component of the velocity vector. In general, we identify a complete vector by a letter with an arrow on top, The magnitude of vector is just A without the arrow, =. What is a vector? A 3- D vector is a mathematical object that has three independent components with laws that distinguish them as vectors. Vectors obey laws of addition, multiplication, and coordinate transformations such as rotations. Given two vectors and we can express the law of vector addition geometrically. + = + Fig. 1 Geometrical definition of vector addition We fix one vector say, we bring vector to vector in such a way that the direction and magnitude of are unchanged (this is called parallel transport). We place the tail of vector at the tip of the arrow of vector. The vector sum is the vector with a tail at the tail of vector and with the arrow at the arrow of vector, as we can see in Fig. 1. If we would like to add more vectors we follow the same procedure until we have included every vector in any order. The analytical approach to add vectors is easiest in Cartesian coordinates. The 2- D case is best for showing the approach. We define a xy- coordinate system. We bring vector so that its tail is at the origin. We draw a right triangle that with one of its sides on the x- axis or y- axis. Out of the sides of the right triangle we made two vectors and in such a way that vector is the sum = +
Fig. 2 Using definition of vector addition we break vector into components = +. Now, we introduce the useful concept of unit vector. A unit vector is simple a vector of length one unit in a given direction. We notice that instead of an arrow we use a hat on top of the letter assigned to the vector. For instance, we call the vector of length one along the positive x- axis. In the same way we call the vector of length one along the positive y- axis and the vector of length one along the positive z- axis Fig. 3 The set of positive x, y and z axes and their corresponding unit vectors Using unit vectors we can write vector = and vector =. Thus, we now can write = + All other vectors are broken in the same way; for instance, = +. The sum of vectors is now simply + = + + +
Vector in component form is the preferred way to report them. The alternative way to describe a vector using magnitude and angle is less useful and can be confusing. If we are required to give magnitude and angle we must include a sketch of the vector, the angle and the coordinate system. Example: Given the vectors : = 10 and angle 50 degrees with the positive x- axis, : = 15 and angle 120 degrees with the positive x- axis, and : = 6 and angle 240 degrees with the positive x- axis find their vector sum = + +. First we form right triangles with for every vector with either x- axis or y- axis. In general we do not use angles grater than 90 degrees as they make things confusing. It is better to use the sign convention for the axis: +x is to the right, +y is up and +z is out of the page. 60 50 30 Fig. 4 We break vectors into components using right triangles We use trigonometry to find the sides of each triangle and write the components = 10 cos 50 = 6.43; = 10 sin 50 = 7.66 = 15cos 60 = 7.5; = 15 sin 60 = 12.99 = 6 sin 30 = 3; = 6 cos 60 = 5.2 We notice that the signs of the components are place by hand by looking at the side of the x- axis and y- axis where the components are.
(link for the sign convention). The components of the vector sum are just the regular addition of x and y components: = 4.07; = 15.45 Dot product The dot product is a scalar. A scalar is a number that does not change under common coordinates transformations such as rotations. An example of a scalar is the length of a vector which does not change when the coordinates are rotated. Given vectors and the dot product is = " cos " where C and D are the magnitudes of the vectors and " is the angle, smaller than 180, between the vectors. It is useful to know the dot product between unit vectors; for instance, = 1 1 cos 90 = 0 = 1 1 cos 0 = 1 We summarize the results = = = 0 = = = 1 As an example let us compute the dot product given = 3 2 and = 4 + 2 = 3 2 4 + 2 = 3 12 + 6 2 + 8 4 Using the table above we get = 12 4 = 16. We would like to find the angle between to and using the dot product definition = " cos " We use Pythagoras theorem to calculate the magnitudes given the vectors in regular coordinates; so we get = 13; = 21. We put all the pieces together and we get 16 = 13 21 cos " We use the inverse cosine and get " = 165.55 Cross Product Given vectors and we define the cross product as
= which is a new vector with the following properties: The magnitude of is given by = " sin " We note that the angle " is less than 180 degrees so the magnitude is always positive, as it should be. The direction of is along a line perpendicular to the plane formed by vectors and. Along this perpendicular line, the direction of is given by the right hand rule: we place vectors and with their tails at a common origin. We bring our right hand to that origin. We keep our four fingers together and the thumb at 90 degrees. On the plane expanded by the vectors, we turn our four fingers from the first vector () towards the second vector () along the angle ". The direction of the thumb is the correct direction of. " Fig. 5 Right had rule: the four fingers of our right hand move from to ; the thumb is along. As an example let us compute the cross product of unit vectors. First the magnitude = 1 1 sin 90 = 1 Since the vectors and expand the xy- plane we know that the direction of the cross product is along the z- axis, perpendicular to this plane. We place our right hand at the origin and when our four fingers move from the positive x- axis to the positive y- axis our thumb points along the positive z- axis; thus, we conclude that
= Similar calculations lead us to conclude that = ; = ; = We note that the order is important; if we reverse the order we get the negative = Finally, the cross product of any vector with itself is always zero = = = 0 As an example let us compute the cross product = given = 3 2 and = 4 + 2 = 3 2 4 + 2 = 3 12 + 6 2 + 8 4 Using the table above we get = = 3 + 6 2 8 = 2 2 3 To check our answer we may verify that the is perpendicular (dot product is zero) to and = 2 2 3 3 2 = 6 + 6 = 0 = 2 2 3 4 + 2 = 2 + 8 6 = 0 We would like to find the angle between to and using the definition of cross product magnitude = " sin " We use Pythagoras theorem to calculate the magnitude of vector, = 17. Thus, we get sin " = 17 13 21 Using the inverse sine we get " = 14,45 Unfortunately, this is not the answer we got using the dot product " = 165,55. The problem is that the sine function gives the exact same value for 14,45 and 165,55 and most calculators will just give you the first answer. Therefore, the best
way to avoid complications when finding the angle between two vectors is using the dot product. There are other ways to calculate the cross product of two vectors, =. A common approach is to calculate the determinant of the array = = + + BASIC VECTOR IDENTITIES = = = To show that these identities are correct we use Cartesian coordinates = + + for all the vectors involved. Next, we write all the terms on each side of the equation and make sure that they appear on both sides. In cases where the expressions are vectors, such as the second identity, we only need to show that one of the three components (x, y, or z) works. Exercises: 1. 2. 3. 4. 5. Given the vectors : = 8 and angle 20 degrees with the positive x- axis, : = 5 and angle 150 degrees with the positive x- axis, and : = 10 and angle 250 degrees with the positive x- axis (a) find their vector sum = + +. (b) Find the angle between + and. (c) Find the cross product. Given the vectors : = 5 and angle 60 degrees with the positive x- axis, : = 10 and angle 120 degrees with the positive x- axis, and : = 6 and angle 290 degrees with the positive x- axis find (a) their vector sum = + +. (b) Find the angle between and. (c) Find the cross product. Given the vectors : = 8 and angle 20 degrees with the positive x- axis, : = 5 and angle 150 degrees with the positive x- axis, and : = 10 and angle 250 degrees with the positive x- axis find (a) their vector sum = + +. (b) Find the angle between + and. (c) Find the cross product. Verify the identity = = Verify the identity = Note: only one component is enough.