Vectors. May Mass. Velocity. Temperature. Distance. Density. Force. Acceleration. Volume

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1 Vectors May Vectors are mathematical qantities that hae direction and magnitde, and can be pictred as arrows. This is in contrast to scalars, which are qantities that hae a nmerical ale bt no direction. 1. Identify the following qantities as either ectors or scalars. Mass Velocity Temperatre Distance Density Force Acceleration Volme 1

2 Adding ectors on a line. Harry Potter rides on a train that is moing with a elocity!. Harry moes with a elocity! with respect to the train. Harry moes in the same direction that the train moes, and the ectors for Harry s and the train s moements are shown below. Harry then casts the inisibility spell on the train. Draw the ector for Harry s moement as seen by a iewer standing otside of the train. Denote this ector by! w.(hint: think abot how large the ector needs to be, and which direction it needs to go in). 3. The rier flows with a elocity!. A boat moes on the rier with a elocity! with respect to the rier (in the direction of the rier). The two ectors are shown below. Draw the ector that shows the boat s elocity with respect to someone standing on the shore of the rier.. A passenger in a plane walks from the front to the back of the plane with a elocity!, while the plane flies in one direction with a elocity!. Draw the ector showing the passenger s elocity with respect to an otside obserer.

3 Adding ectors on the plane When talking abot ectors, it is common to refer to the end with the arrow as the head, and the starting point as the tail. When adding ectors! a and! b,thefirst step is to moe the ectors in sch a way that the head of! a toches the tail of! b. Then, yo connect the tail of! a to the head of! b. Notice that this is what yo did when adding ectors on a line. Below is an example showing the addition of ectors! a and! b,resltinginector! c. This is often referred to as head-to-tail addition method becase yo are ptting the head of the second ector (! b )tothetailofthe first ector (! a ). c b a 5. Add p the following ectors. (a) (b) (c) 3

4 (d). In the example below, redraw the ectors in sch a way that the tail of the horizontal ector is attached to the head of the ertical one. Then add the ectors. Now repeat the problem by drawing the tail of the ertical ector attached to the head of the horizontal one. Is yor answer different? 7. Add p the following ectors. (a) (b)

5 (c). Yo are walking with a elocity! on a boat which is moing with a elocity!,as shown below. Draw the ector showing yor elocity with respect to an otside iewer standing on the shore. 9. The train Harry rides on is a steam train, and a lot of steam is emitted to the atmosphere. The steam leaes pward with a elocity! while the train moes horizontally with a elocity!. Draw the ector that shows the elocity of the steam with respect to an otside iewer. 5

6 Sbtracting ectors on the plane Recall that sbtraction is the operation opposite to addition. For nmbers, performing sbtraction a b + c = a. Forexample b means finding sch a nmber c that 5 3= becase 3+=5. Similarly, finding!! means finding a ector! w sch that! =! +! w.below are two drawings, one where! w =!!, and one where! w =!!. Circle the one where! w =!!. w w 1. Sbtract the ectors shown below. That is, draw the ector eqal to!!. (a) (b)

7 (c) (d) 11. Yo can also sbtract ectors!! by reersing the direction of! and then performing an addition problem. This is similar to sbtracting scalars on a nmber line, becase sbtraction on a nmber line is the same as addition of a negatie of a nmber. For example, 5 3 = 5+( 3) ( ) = + ( ( )) = + Try both methods on the ectors below, and write down which method yo prefer. 7

8 1. Yo are walking on a plane. On the pictre below, ector! shows yor elocity with respect to an otside iewer and ector! shows yor elocity with respect to the plane. Show the elocity of the plane with respect to an otside iewer. 13. Yo are walking on a boat, where ector! below shows yor elocity with respect to an otside iewer and ector! shows the elocity of the boat with respect to an otside iewer. Show the yor elocity with respect to the boat.

9 Vectors on a coordinate plane It is common to draw ectors on a coordinate plane. Similar to how a point is denoted by (x, y), aectorcanbespecifiedbyitscoordinates<x,y>.aector<, 5 > moes in the x-direction by nits and the y-direction by 5 nits. Note that it is cstomary to draw ectors starting at the origin, bt this does not always hae to be the case. 1. Below is the ector <, 5 >. Drawector< 5, > by starting at point (, 1). y x 15. Below are three ectors on the coordinate plane. Write down the components of each ector sing <x,y>notation. y (a) (b) (c) x 9

10 1. Using the coordinate plane below, draw the ector that reslts from adding ector < 3, > with ector <, >. (Hint: first draw ector < 3, > starting at the origin). y x What is the reslting ector, sing the notation <x,y>? 17. Use the coordinate plane below to draw the ector reslting from < 1, > + <, 1 >. y x What is the reslting ector, sing the form <x,y>? 1

11 1. Use the coordinate plane below to draw the ector reslting from < 7, 9 > < 5, 1 >. y x What is the reslting ector, sing the form <x,y>? 19. Use the coordinate plane below to draw the ector reslting from < 9, > < 7, >. y x What is the reslting ector, sing the form <x,y>? 11

12 . Using yor answers from the preios problems, determine the ectors that reslt from the following operations withot drawing the ectors. (a) < 15, > + < 5, 1 >= (b) < 9, 1 > < 1, 9 >= (c) < a, b > + <c,d>= (d) < a, b > <c,d>= 1