Vectors A Vector has Two properties Magnitude and Direction. That s a weirder concept than you think. A Vector does not necessarily start at a given point, but can float about, but still be the SAME vector. A vector is usually denoted in bold, like vector a, or sometimes it is denoted a, or many other deviations exist in various text books. How do your Draw a vector? When you draw a Vector, its just like an Arrow. We say that a vector has a head and a tail. I am hoping you can work out which end is which J Addition of vectors: When we add vectors geometrically we simply align them head to tail, (like elephants). The resultant Vector is drawn from the tail of the first vector, to the head of the second vector. Or we can add multiple vectors together. TASK: Draw 2 vectors, a and b anywhere on your page. Now; Geometrically add a + b Geometrically add b + a What is your hypothesis about the order in which you add vectors? TASK: Draw any vector and label it a. Now, geometrically draw vector a. Now, draw another vector a somewhere else on your page What is the difference between your 2 vectors of a? 1
TASK: Draw any vector on your page and label it c. Now; Try to draw a vector that represents 2c. TASK: Draw any vector on your page and label it d. Now; If I say that Vector division is not defined, think about what you could do to this vector to make it smaller, but without changing its direction? So, you were right you can add and subtract vectors in any order. It doesn t matter. You can represent a vector geometrically ANYWHERE on your page as it does not have a Start point. You can also SCALAR multiply a vector. And if the Scalar is less than 1, then the vector gets smaller. Lets just double check some things: Can you see that vector d is EXACTLY the SAME Vector as b 2
Consider the following geometric representation of vector addition. Looks pretty weird / abstract. Can you show more clearly the representation of a + b Does this look better? 3
It is important that you use your eyes to help your brain see these operations as easy. Maths is about seeing through the complexity, and looking at things strategically to make it more easy to understand. Note that I used the origin of a Cartesian plane as my start point. Although vectors do not have a start point, sometimes this makes it easier to see what is happening! 4
How about a Scalar of a vector. Although we denote this operation as ka (k is a constant) try NOT to think of it as Multiplication. What is a scalar? it s the enlargement, or reduction of the magnitude of something. A scale model car, is supposed to be an exact replica of a real car, just shrunk down. The word Scalar comes up again later and we Need to think of Scalar differently than Multiplication? Do Exercise 7A 5
Position vectors in 2 and 3 Dimensions Just drawing vectors is a little general, with limited accuracy. It is convenient to represent vectors by components in the x and y direction and the x direction is denoted with an 0 and the y direction is denoted with a 1. (the third Cartesian dimension we call z, and in vectors this aligns to k) In 2D Cartesian form: a = xı + yȷ In 3D Cartesian form: a = xı + yȷ + zk Sometimes we abbreviate it even further and we can simply put them in pointy brackets. In 2D Cartesian form: a = x, y In 3D Cartesian form: a = x, y, z Vectors can be scalar multiplied, added and subtracted. As you have seen we can do this geometrically, but these operations can also be performed mathematically. But first, lets work out how we get x and y. Here we use Trigonometry. Because vectors have a magnitude and direction, we could simply say vector is 10 units long and has an angle of 30 degrees. ** All angles are measured from the positive x axis, in an anti-clockwise direction.** OK, think Yr 9 trigonometry The length of the vector is the Hypotenuse. The size of the x component uses the cosine ratio and the length of the y component uses the sine ratio. It s a simple as: x = vector length cos θ y = vector length sin θ Hang on, lets talk in mathematical terms J 6
The length of the vector is called Magnitude, and is denoted by a. *** caution, this is the same symbol we used for the determinant of a matrix, so make sure you don t get confused! So lets say x = a cos θ y = a sin θ But what if we don t know the magnitude of the vector? Using Pythagoras, we have a = x E + y E But, what if we don t know the angle what if we don t know θ. Hopefully you will be thinking about the Tangent ratio? θ = tan FG y x 7
OK lets put it all together. As different text books say things differently, there will be many different representations of the same concept. Lets look at just a few: In 2D: Given Vector OP = a = a = xı + yȷ = a cos θ ı + a sin θ ȷ = x, y, We have a Magnitude, a = x E + y E And the direction of the Vector is given by θ, where tan θ = K L In 3D: Given Vector OP = a = a = xı + yȷ + zk = x, y, z, a = x E + y E + z E OK, have you noticed that the i, j & k are wearing hats. The hat tells us that the vector is a special vector called a Unit Vector. So we need to talk about Unit vectors. A unit vector is a vector of Magnitude 1 unit. Think about a stick 10 units long, to get that stick 1 unit long, we would scalar it! Vectors are the same. We need to Scale our original vector, either bigger, by G GP or smaller so that its magnitude is 1. Hence we have; a = 1 a a And we shall simplify it to read as; a = a a Now, specifically thinking about i, j & k, these are Unit Vectors in the direction of the x, y & z axis respectfully. So, hopefully you can see how the representation works a = xı + yȷ The vector a is actually the same as the geometrical representation of the sum of the two vectors represented by xı and yȷ J 8
Something further to confuse you. There is a difference between a Position Vector, and a Vector. A Position Vector describes a Position in the plane, or Point. Whereas a Vector is seen as a line that has magnitude and direction (but no starting point). Lets do a Simple example Consider the Position Vectors 0,1 and 2,3 hang on, don t just consider them, plot them on a Cartesian plane. These two Position Vectors describe a Vector that joins these 2 points. Can you work out the Vector that joins these 2 points? The answer, v = 2ı + 2ȷ Make sure you can see how this works it is important!!! Anyway Lets get to some Maths!!! Adding is as easy as: Given a = 3ı 4ȷ + 3k, and b = 2ı + 3ȷ + 2k a + b = 3 + 2 ı + 4 + 3 ȷ + 3 + 2 k = 5ı ȷ + 5k Subtracting is the same. Doing is learning. Humans learn by doing. Practice is everything, so go Practice! Do Exercise 7B 9
TASK: On a Cartesian plane, plot the coordinates 1, 1, 3, 1, and 2, 3. What shape is this? We could also think of these as Position Vectors, as Position Vectors represent Points, so I could have said, plot the Position Vectors 1, 1, 3, 1 and 2, 3 and you would have drawn the exact same diagram. Hopefully you said Triangle, and specifically an Isosceles Triangle. Not only do three points represent a triangle, but a triangle also has 3 sides. Rather than describing the triangle with Position Vectors (points), we can also describe a triangle with three Vectors. TASK: Given the three points of the previous task, use your thorough understanding of how to geometrically arrange vectors, to come up with the 3 vectors that describe the triangle in the previous Task. Lets allocate the Position Vectors as: A = 1, 1, B = 3, 1, C = 2, 3 and ask you to find Vectors AB, BC and AC that describe that same triangle. Answer to task: AB = A + B = 1, 1 + 2, 3 = 1, 2 BC = B + C = 2, 3 + 3, 1 = 1, 2 AC = A + C = 1, 1 + 3, 1 = 2, 0 10
The Dot Product (Scalar Product) A dot product is a Scalar Value that is the result of an operation of two vectors with the same number of components. It is the Sum of the Products of each respective component. a b = x G x E + y G y E + z G z E Geometrically speaking, the Dot Product refers to multiplying the magnitude of one vector, with the magnitude of the Component of the other vector that is in the same direction as it. So we have another formula for it. a b = a b cosθ Hence, we can say: a b = a b cosθ = x G x E + y G y E + z G z E We use this rule to Find the angle between two vectors in 2D and 3D. The Dot Product is a good way to see if vectors are perpendicular, because cos 90 = 0, the dot product will be equal to Zero. Do Exercise 7C If you look on the interweb, you may find many confusing explanations of the dot product. This website is as clear as I can find and does an adequate job at trying to explain it! http://mathinsight.org/image/dot_product_projection 11
Resolutes There are three formula s to remember Remember: Scalar Resolute of v on u is given by Vector Resolute of v parallel to u is Vector Resolute of v perpendicular to u is v u v = v u u v f = v v u u *** Note: the way I write the scalar resolute is different than the book. By writing it MY way, it aligns to the way you say it. The resolute of v on u so the v comes first *** but I suggest you also Understand how it works from first principles. The Scalar Resolute: This is how to see the Scalar resolute of vector A on Vector B. 12
It is simply the Magnitude of the component of Vector A in the direction of vector B. It s as simple as that. To avoid having to work out θ and A, it is quick and convenient to recall the formula Scalar Resolute = v u but understanding it from geometrically is also advantageous for you. Scalar means Number, so a scalar resolute is a Number. The parallel Vector Resolute: I can use the exact same diagram The Vector resolute of A, onto B, is simply the vector component of A, in the direction of B. So if we have a Unit Vector for B, we simply multiply it by the scalar resolute and we end up with A cos θ B ** Note: in the above, B is a Unit VECTOR. Or in formula form we have: v = v u u Hence, the result of this operation is a Vector. A scalar multiplied by a unit vector, is a Vector! Specifically, it is the component vector of A in the direction of B. Vector Resolute as the name implies, your answer has to be a VECTOR. 13
The Perpendicular Vector Resolute: TASK: Draw the same diagram in your book. Draw the dotted line in Red (with an arrow at the top). This is the component of A that is perpendicular to B. Can you show Geometrically, that this can be represented by vector A minus the parallel vector resolute, or; We can show this geometrical representation mathematically as; A A cos θ B or in formula form: v f = v v u u Do Exercise 7D Can you trust the Text Book Answers I think they have done Question 4 incorrectly, what do you think? Chapter 7E Time Varying Vectors Although you may be glad we don t have to know this chapter, ask me why you should be disappointed J 14