Blue and purple vectors have same magnitude and direction so they are equal. Blue and green vectors have same direction but different magnitude.

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2 A ector is a quantity that has both magnitude and direction. It is represented by an arrow. The length of the ector represents the magnitude and the arrow indicates the direction of the ector. Blue and orange ectors hae same magnitude but different direction. Blue and purple ectors hae same magnitude and direction so they are equal. Blue and green ectors hae same direction but different magnitude. Two ectors are equal if they hae the same direction and magnitude (length).

3 How can we find the magnitude if we hae the initial point and the terminal point? The distance formula Initial Point ( ) x 1, y 1 P magnitude is the length direction is this angle Q ( ) x 2, y 2 Terminal Point 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!)

4 Although it is possible to do this for any initial and terminal points, since ectors are equal as long as the direction and magnitude are the same, it is easiest to find a ector with initial point at the origin and terminal point (x, y). Initial Point ( 0, 0 ) x 1, y 1 P direction is this angle Q ( x, y ) x 2, y 2 Terminal Point A ector whose initial point is the origin is called a position ector If we subtract the initial point from the terminal point, we will hae an equialent ector with initial point at the origin.

5 To To add ectors, we put the initial point of of the second ector on on the terminal point of of the first ector. The resultant ector has an an initial point at at the initial point of of the first ector and a terminal point at at the terminal point of of the second ector (see below--better shown than put in in words). Terminal point of w + w w w Initial point of Moe w oer keeping the magnitude and direction the same.

6 The negatie of a ector is ust a ector going the opposite way. A number multiplied in front of a ector is called a scalar. It means to take the ector and add together that many times. 3

7 u w 3w w w w Using the ectors shown, find the following: u + u u u 2u + 3w + u u w w w

8 Vectors are denoted with bold letters = a b i = ai + b a = b (a, b) 3 = i 2 i i This is the notation for a position ector. This means the point (a, b) is the terminal point and the initial point is the origin. We use ectors that are only 1 unit long to build position ectors. i is a ector 1 unit long in the x direction and is a ector 1 unit long in the y direction. (3, 2) = 3 i + 2

9 If we want to add ectors that are in the form ai + b, we can ust add the i components and then the components. = 2 i + 5 w = 3i 4 + w = 2i i 4 Let's look at this geometrically: Can you see from this picture how to find the length of? 5 3i 2i w i 4 = i + When we want to know the magnitude of the ector (remember this is the length) we denote it ( ) ( ) 2 = = 29

10 A unit ector is a ector with magnitude 1. If we want to find the unit ector haing the same direction as a gien ector, we find the magnitude of the ector and diide the ector by that alue. w = 3i ( ) ( ) What is w? w = = 25 = 5 If we want to find the unit ector haing the same direction as w we need to diide w by 5. u 3 4 Let's check this to see if it really is = i 1 unit long u = + = =

11 If we know the magnitude and direction of the ector, let's see if we can express the ector in ai + b form = 5, α = 150 As usual we can use the trig we know to find the length in the horizontal direction and in the ertical direction. ( cosα sinα ) = i+ = ( cos150 i + sin150 ) = i 5 2

12 Acknowledgement I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint. Shawna has kindly gien permission for this resource to be downloaded from and for it to be modified to suit the Western Australian Mathematics Curriculum. Stephen Corcoran Head of Mathematics St Stephen s School Carramar