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A vector is a quantity that has both magnitude and direction. It is represented by an arrow. The length of the vector represents the magnitude and the arrow indicates the direction of the vector. Blue and orange vectors have same magnitude but different direction. Blue and purple vectors have same magnitude and direction so they are equal. Blue and green vectors have same direction but different magnitude. Two vectors are equal if they have the same direction and magnitude (length). How can we find the magnitude if we have the initial point and the terminal point? The distance formula Q x2 , y 2 Terminal Point Initial Point x1 , y1 P How can we find the direction? (Is this all looking familiar for each application? You can make a right triangle and use trig to get the angle!) Although it is possible to do this for any initial and terminal points, since vectors are equal as long as the direction and magnitude are the same, it is easiest to find a vector with initial point at the origin and terminal point (x, y). Initial Point Q xx, 2 , yy 2 Terminal Point A vector whose initial point is the origin is called a position vector 0x1,, 0y1 P If we subtract the initial point from the terminal point, we will have an equivalent vector with initial point at the origin. To Toadd addvectors, vectors,we weput putthe theinitial initialpoint pointof ofthe thesecond second vector vector on onthe theterminal terminalpoint pointof ofthe thefirst firstvector. vector. The The resultant resultantvector vector has hasan aninitial initialpoint pointat atthe theinitial initialpoint point of ofthe thefirst firstvector vector and andaaterminal terminalpoint pointat atthe theterminal terminal point pointof ofthe thesecond secondvector vector(see (seebelow--better below--bettershown shown than thanput putin inwords). words). Terminal point of w vw Initial point of v v w w Move w over keeping the magnitude and direction the same. The negative of a vector is just a vector going the opposite way. v v A number multiplied in front of a vector is called a scalar. It means to take the vector and add together that many times. 3v v v v v u w Using the vectors shown, find the following: u v 3w w w w uv u 2u 3w v v u u u v w w w v Vectors Worksheet #1 Head-Minus Tail Rule Prove that the two vectors RS and PQ are equivalent. 1.) R = (-4, 7) S = (-1, 5) P = (0, 0) Q = (3, -2) 2.) R = (7, -3) S = (4, -5) P = (0, 0) Q = (-3, -2) 3.) R = (2, 1) S = (0, -1) P = (1, 4) Q = (-1, 2) 4.) R = (-2, -1) S = (2, 4) P = (-3, -1) Q = (1, 4) Vectors are denoted with bold letters This is the notation for a position vector. This means the point (a, b) is the terminal point and the initial point is the origin. a v a b v ai bj We use vectors that are only 1 b unit long to build position (a, b) j i 3 v 2 vectors. i is a vector 1 unit long in the x direction and j is a vector 1 unit long in the y direction. (3, 2) j j i i i v 3i 2 j If we want to add vectors that are in the form ai + bj, we can just add the i components and then the j components. v 2i 5j w 3i 4 j v w 2i 5j 3i 4 j i j Let's look at this geometrically: Can you see from this picture 5j how to find the length of v? 3i w v 2i i 4j j When we want to know the magnitude of the vector (remember this is the length) we denote it v 2 5 2 29 2 Vectors Worksheet #2 Using the following vectors: P = (-2, 2) Q = (3, 4) R = (-2, 5) S = (2, -8) Find: PQ RS QR PS 2QS (√2)PR 3QR + PS PS – 3PQ Vector Worksheet #3 Performing Vector Operations Let u = <-1, 3> and v = <4, 7> Find the component form of the following vectors. u+v 3u 2u + (-1)v u + v = <-1, 3> + <4, 7> = <-1 + 4, 3 + 7> = <3, 10> 3u = 3<-1, 3> = <-3, 9> 2u + (-1)v = 2<-1, 3> + (-1)<4, 7> = <-2, 6> + <-4, -7> = <-6, -1> Performing Vector Operations Now it’s your turn! Let u = <-1, 3> , v = <2, 4> and w = <2, -5> Find: u+v u + (-1)v u–w 3v 2u + 3w 2u – 4v – 2u – 3v –u–v Performing Vector Operations Now it’s your turn! Let u = <-1, 3> , v = <2, 4> and w = <2, -5> Find: u+v u + (-1)v u–w 3v 2u + 3w 2u – 4v – 2u – 3v –u–v = = = = = = = = <1, 7> <-3, -1> <-3, 8> <6, 12> <4, -9> <-10, -10> <-4, -18> <-1, -7> Unit Vectors and Direction Angles • Any vector can be broken down into its components: a horizontal component and a vertical component. • In addition, any vector can be written as an expression in terms of a standard unit vector. • Unit vectors help us separate vectors into components—a scalar and a unit vector. Unit Vectors and Direction Angles • • • • The standard unit vectors are i and j. i = <1, 0> j = <0, 1> Using this style, we can now express vectors as a linear combination. Unit Vectors and Direction Angles • Vector v = <a, b> = <a, 0> + <0, b> = a<1, 0> + b<0, 1> • We now have linear combination. v = a•i + b•j Unit Vectors and Direction Angles In vector v = a•i + b•j a and b are now scalars and express the horizontal and vertical components of vector v. We can use Trigonometry to calculate a direction angle for our vector. A unit vector is a vector with magnitude 1. If we want to find the unit vector having the same direction as a given vector, we find the magnitude of the vector and divide the vector by that value. w 3i 4 j w 3 4 2 What is w ? 2 25 5 If we want to find the unit vector having the same direction as w we need to divide w by 5. 3 4 u i j 5 5 Let's check this to see if it really is 1 unit long. 2 2 25 3 4 u 1 25 5 5 If we know the magnitude and direction of the vector, let's see if we can express the vector in ai + bj form. v 5, 150 5 As usual we can use the trig we know to find the length in the horizontal direction and in the vertical direction. 150 v v cosi sin j 5 3 5 v 5cos150i sin 150 j i j 2 2 Acknowledgement I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint. www.slcc.edu Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum. Stephen Corcoran Head of Mathematics St Stephen’s School – Carramar www.ststephens.wa.edu.au