Have you ever arrived 15 minutes late for a movie?  Even though you may eventually learn what is happening in the plot, you know that you're not having the same experience that you could have had you seen the beginning.  If the movie had an intriguing plot, the effects of missing the beginning can be worse and you may never grasp the entire meaning of the film.

Module 1 is much like the beginning of a movie with a complex plot. If you miss the main ideas at this point, you may struggle with some of the rest of the course. Although the ideas are meant to be a review of Physics 20, you will have to pay close attention to the details of how to communicate vector solutions properly. Learning how to communicate your solutions in a way that will help you be successful in the whole course, including the diploma exam, is important. Thus, as you review vectors, be sure to check your answers closely with the solutions provided.

 

Read Vector Methods in One Dimension
Begin your review of vectors by reading the introduction and the section titled "Vector Methods in One-Dimension" on pages 70 to 75 of the textbook.

 What is a resultant vector?

A resultant is a single vector that represents the sum of two or more other vectors.


How is a resultant vector drawn?

The resultant is shown graphically by an arrow that connects the tail of the first vector with the head of the last vector.


Does order matter when you add vectors together?

No, the order of adding vectors has no effect on the resultant.



Read Graphical Method of Adding Vectors
To review how vectors are added graphically in two dimensions, read from the bottom of page 76 to page 81 of your textbook.

The angle between two vectors does not always have to be a right angle. For example, the two vectors could be arranged as shown in the following diagram.



Use graphical methods to determine the resultant of these two vectors.


Scale: Let 1 cm = 10 N



At this point, the approach that been using is mainly graphical-solutions could be obtained by drawing the vectors with a ruler and a protractor.  When a more precise answer is required an alternative mathematical approach is used.



Read Proper Use of Vector Notation
To review how a more precise method of determining vectors, read pages 83 to 89 in the text. Note the use of notations in "Example 2.5" on page 88.

The following guidelines explain the proper use of vector notation in Physics 30:

  • You should be able to add and subtract vector quantities from each other. You should also be able to multiply and divide vector quantities by constants or by scalar quantities.
  • You are not expected to multiply and divide vector quantities by other vector quantities. In this course it is customary to drop the vector notation when multiplying or dividing vectors because it is beyond the scope of this course.

Solve question 11 on page 90 of your textbook.


Given: 

 «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mover»«mi»d«/mi»«mo»§#8594;«/mo»«/mover»«mn»1«/mn»«/msub»«mo»=«/mo»«mn»45«/mn»«mo».«/mo»«mn»0«/mn»«mo»§#160;«/mo»«mi»m«/mi»«mo»§#160;«/mo»«mo»[«/mo»«msup»«mn»310«/mn»«mo»§#8728;«/mo»«/msup»«mo»]«/mo»«mspace linebreak=¨newline¨»«/mspace»«msub»«mover»«mi»d«/mi»«mo»§#8594;«/mo»«/mover»«mn»2«/mn»«/msub»«mo»=«/mo»«mn»35«/mn»«mo».«/mo»«mn»0«/mn»«mo»§#160;«/mo»«mi»m«/mi»«mo»§#160;«/mo»«mo»[«/mo»«msup»«mn»135«/mn»«mo»§#8728;«/mo»«/msup»«mo»]«/mo»«/math»


Required:

The resultant displacement, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mover»«mi»d«/mi»«mo»§#8594;«/mo»«/mover»«mi»R«/mi»«/msub»«/math»

Analysis and Solution:

Step 1: These vectors are not collinear, so find the x- and y-components of each one:



A vector component addition diagram showing the x1 component in the positive x-direction and the y1 component in the negative y-direction being added with a resultant displacement at +310o

 

A vector component addition diagram showing the resultant x component in the positive x direction and the resultant y component in the negative y direction being added to show the resultant displacement.


Note: Vector notation is not used for the components because they are scalars.

Displacementx-componenty-component
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mover»«mi»d«/mi»«mo»§#8594;«/mo»«/mover»«mn»1«/mn»«/msub»«mo»=«/mo»«mn»45«/mn»«mo».«/mo»«mn»0«/mn»«mo»§#160;«/mo»«mi»m«/mi»«mo»§#160;«/mo»«mo»[«/mo»«msup»«mn»310«/mn»«mo»§#8728;«/mo»«/msup»«mo»]«/mo»«/math»
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd»«msub»«mi»d«/mi»«mrow»«mn»1«/mn»«mi»x«/mi»«/mrow»«/msub»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msub»«mover»«mi»d«/mi»«mo»§#8594;«/mo»«/mover»«mn»1«/mn»«/msub»«mi»cos«/mi»«mi»§#952;«/mi»«/mtd»«/mtr»«mtr»«mtd»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»(«/mo»«mn»45«/mn»«mo».«/mo»«mn»0«/mn»«mi»m«/mi»«mo»)«/mo»«mi»cos«/mi»«mo»(«/mo»«msup»«mn»50«/mn»«mo»§#8728;«/mo»«/msup»«mo»)«/mo»«/mtd»«/mtr»«mtr»«mtd»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»28«/mn»«mo».«/mo»«mn»93«/mn»«mo»§#160;«/mo»«mi»m«/mi»«/mtd»«/mtr»«/mtable»«/math»
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd»«msub»«mi»d«/mi»«mrow»«mn»1«/mn»«mi»y«/mi»«/mrow»«/msub»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msub»«mover»«mi»d«/mi»«mo»§#8594;«/mo»«/mover»«mn»1«/mn»«/msub»«mi»sin«/mi»«mi»§#952;«/mi»«/mtd»«/mtr»«mtr»«mtd»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mrow»«mo»-«/mo»«mo»(«/mo»«mn»45«/mn»«mo».«/mo»«mn»0«/mn»«mi»m«/mi»«mo»)«/mo»«mi»sin«/mi»«mo»(«/mo»«msup»«mn»50«/mn»«mo»§#8728;«/mo»«/msup»«mo»)«/mo»«/mrow»«/mtd»«/mtr»«mtr»«mtd»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»-«/mo»«mn»34«/mn»«mo».«/mo»«mn»47«/mn»«mo»§#160;«/mo»«mi»m«/mi»«/mtd»«/mtr»«/mtable»«/math»
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mover»«mi»d«/mi»«mo»§#8594;«/mo»«/mover»«mn»2«/mn»«/msub»«mo»=«/mo»«mn»35«/mn»«mo».«/mo»«mn»0«/mn»«mo»§#160;«/mo»«mi»m«/mi»«mo»§#160;«/mo»«mo»[«/mo»«msup»«mn»135«/mn»«mo»§#8728;«/mo»«/msup»«mo»]«/mo»«/math»
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd»«msub»«mi»d«/mi»«mrow»«mn»2«/mn»«mi»x«/mi»«/mrow»«/msub»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msub»«mover»«mi»d«/mi»«mo»§#8594;«/mo»«/mover»«mn»2«/mn»«/msub»«mi»cos«/mi»«mi»§#952;«/mi»«/mtd»«/mtr»«mtr»«mtd»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mrow»«mo»-«/mo»«mo»(«/mo»«mn»35«/mn»«mo».«/mo»«mn»0«/mn»«mi»m«/mi»«mo»)«/mo»«mi»cos«/mi»«mo»(«/mo»«msup»«mn»45«/mn»«mo»§#8728;«/mo»«/msup»«mo»)«/mo»«/mrow»«/mtd»«/mtr»«mtr»«mtd»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»-«/mo»«mn»24«/mn»«mo».«/mo»«mn»75«/mn»«mo»§#160;«/mo»«mi»m«/mi»«/mtd»«/mtr»«/mtable»«/math»
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd»«msub»«mi»d«/mi»«mrow»«mn»2«/mn»«mi»y«/mi»«/mrow»«/msub»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msub»«mover»«mi»d«/mi»«mo»§#8594;«/mo»«/mover»«mn»2«/mn»«/msub»«mi»sin«/mi»«mi»§#952;«/mi»«/mtd»«/mtr»«mtr»«mtd»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mrow»«mo»(«/mo»«mn»35«/mn»«mo».«/mo»«mn»0«/mn»«mi»m«/mi»«mo»)«/mo»«mi»cos«/mi»«mo»(«/mo»«msup»«mn»45«/mn»«mo»§#8728;«/mo»«/msup»«mo»)«/mo»«/mrow»«/mtd»«/mtr»«mtr»«mtd»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»24«/mn»«mo».«/mo»«mn»75«/mn»«mo»§#160;«/mo»«mi»m«/mi»«/mtd»«/mtr»«/mtable»«/math»
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mover»«mi»d«/mi»«mo»§#8594;«/mo»«/mover»«mi»R«/mi»«/msub»«/math»«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd»«msub»«mi»d«/mi»«mrow»«mi»R«/mi»«mi»x«/mi»«/mrow»«/msub»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msub»«mi»d«/mi»«mrow»«mn»1«/mn»«mi»x«/mi»«/mrow»«/msub»«mo»+«/mo»«mo»§#160;«/mo»«msub»«mi»d«/mi»«mrow»«mn»2«/mn»«mi»x«/mi»«/mrow»«/msub»«/mtd»«/mtr»«mtr»«mtd»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mrow»«mn»28«/mn»«mo».«/mo»«mn»93«/mn»«mo»§#160;«/mo»«mi»m«/mi»«mo»§#160;«/mo»«mo»+«/mo»«mo»(«/mo»«mo»-«/mo»«mn»24«/mn»«mo».«/mo»«mn»75«/mn»«mo»§#160;«/mo»«mi»m«/mi»«mo»)«/mo»«/mrow»«/mtd»«/mtr»«mtr»«mtd»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»4«/mn»«mo».«/mo»«mn»18«/mn»«mo»§#160;«/mo»«mi»m«/mi»«/mtd»«/mtr»«/mtable»«/math»«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd»«msub»«mi»d«/mi»«mrow»«mi»R«/mi»«mi»y«/mi»«/mrow»«/msub»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msub»«mi»d«/mi»«mrow»«mn»1«/mn»«mi»y«/mi»«/mrow»«/msub»«mo»+«/mo»«mo»§#160;«/mo»«msub»«mi»d«/mi»«mrow»«mn»1«/mn»«mi»y«/mi»«/mrow»«/msub»«/mtd»«/mtr»«mtr»«mtd»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»-«/mo»«mn»34«/mn»«mo».«/mo»«mn»47«/mn»«mo»§#160;«/mo»«mi»m«/mi»«mo»§#160;«/mo»«mo»+«/mo»«mn»24«/mn»«mo».«/mo»«mn»75«/mn»«mo»§#160;«/mo»«mi»m«/mi»«/mtd»«/mtr»«mtr»«mtd»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»-«/mo»«mn»9«/mn»«mo».«/mo»«mn»72«/mn»«mo»§#160;«/mo»«mo»§#160;«/mo»«mi»m«/mi»«/mtd»«/mtr»«/mtable»«/math»


Note: A negative sign was added to d2x because it points in the negative x-direction.
         A negative sign was added to d1y because it points in the negative y-direction.

Step 2: Determine the magnitude of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mover»«mi»d«/mi»«mo»§#8594;«/mo»«/mover»«mi»R«/mi»«/msub»«/math»

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd»«msup»«mi»c«/mi»«mn»2«/mn»«/msup»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mi»a«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mo»§#160;«/mo»«msup»«mi»b«/mi»«mn»2«/mn»«/msup»«/mtd»«/mtr»«mtr»«mtd»«msubsup»«mover»«mi»d«/mi»«mo»§#8594;«/mo»«/mover»«mi»R«/mi»«mn»2«/mn»«/msubsup»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msubsup»«mi»d«/mi»«mrow»«mi»R«/mi»«mi»x«/mi»«/mrow»«mn»2«/mn»«/msubsup»«mo»+«/mo»«msubsup»«mi»d«/mi»«mrow»«mi»R«/mi»«mi»y«/mi»«/mrow»«mn»2«/mn»«/msubsup»«/mtd»«/mtr»«mtr»«mtd»«mfenced open=¨|¨ close=¨|¨»«msub»«mover»«mi»d«/mi»«mo»§#8594;«/mo»«/mover»«mi»R«/mi»«/msub»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msqrt»«msubsup»«mi»d«/mi»«mrow»«mi»R«/mi»«mi»x«/mi»«/mrow»«mn»2«/mn»«/msubsup»«mo»+«/mo»«msubsup»«mi»d«/mi»«mrow»«mi»R«/mi»«mi»y«/mi»«/mrow»«mn»2«/mn»«/msubsup»«/msqrt»«/mtd»«/mtr»«mtr»«mtd»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msqrt»«mo»(«/mo»«mn»4«/mn»«mo».«/mo»«mn»18«/mn»«mo»§#160;«/mo»«mi»m«/mi»«msup»«mo»)«/mo»«mn»2«/mn»«/msup»«mo»+«/mo»«mo»(«/mo»«mo»-«/mo»«mn»9«/mn»«mo».«/mo»«mn»72«/mn»«/msqrt»«mi»m«/mi»«mi»x«/mi»«msup»«mo»)«/mo»«mn»2«/mn»«/msup»«/mtd»«/mtr»«mtr»«mtd»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»10«/mn»«mo».«/mo»«mn»6«/mn»«mo»§#160;«/mo»«mi»m«/mi»«/mtd»«/mtr»«/mtable»«/math»

 


Step 3: Determine the direction of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mover»«mi»d«/mi»«mo»§#8594;«/mo»«/mover»«mi»R«/mi»«/msub»«/math»

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd»«mi»tan«/mi»«mo»§#160;«/mo»«mi»§#952;«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«msub»«mi»d«/mi»«mrow»«mi»R«/mi»«mi»y«/mi»«/mrow»«/msub»«msub»«mi»d«/mi»«mrow»«mi»R«/mi»«mi»x«/mi»«/mrow»«/msub»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mi»tan«/mi»«mrow»«mo»-«/mo»«mn»1«/mn»«/mrow»«/msup»«mfenced open=¨[¨ close=¨]¨»«mfrac»«msub»«mi»d«/mi»«mrow»«mi»R«/mi»«mi»y«/mi»«/mrow»«/msub»«msub»«mi»d«/mi»«mrow»«mi»R«/mi»«mi»x«/mi»«/mrow»«/msub»«/mfrac»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mi»tan«/mi»«mrow»«mo»-«/mo»«mn»1«/mn»«/mrow»«/msup»«mfenced open=¨[¨ close=¨]¨»«mfrac»«mrow»«mn»9«/mn»«mo».«/mo»«mn»72«/mn»«mo»§#160;«/mo»«menclose notation=¨updiagonalstrike¨»«mi»m«/mi»«/menclose»«/mrow»«mrow»«mn»4«/mn»«mo».«/mo»«mn»18«/mn»«mo»§#160;«/mo»«menclose notation=¨updiagonalstrike¨»«mi»m«/mi»«/menclose»«/mrow»«/mfrac»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»66«/mn»«mo».«/mo»«msup»«mn»7«/mn»«mo»§#8728;«/mo»«/msup»«/mtd»«/mtr»«/mtable»«/math»

 


Paraphrase:

The final displacement of the skateboarder is 10.6 m [293°]