<< Chapter < Page Chapter >> Page >

Resultant vectors

  1. Harold walks to school by walking 600 m Northeast and then 500 m N 40 W. Determine his resultant displacement by using accurate scale drawings.
  2. A dove flies from her nest, looking for food for her chick. She flies at a velocity of 2 m · s - 1 on a bearing of 135 and then at a velocity of 1,2 m · s - 1 on a bearing of 230 . Calculate her resultant velocity by using accurate scale drawings.
  3. A squash ball is dropped to the floor with an initial velocity of 2,5 m · s - 1 . It rebounds (comes back up) with a velocity of 0,5 m · s - 1 .
    1. What is the change in velocity of the squash ball?
    2. What is the resultant velocity of the squash ball?

Remember that the technique of addition and subtraction just discussed can only be applied to vectors acting along a straight line. When vectors are not in a straight line, i.e. at an angle to each other, the following method can be used:

A more general algebraic technique

Simple geometric and trigonometric techniques can be used to find resultant vectors.

A man walks 40 m East, then 30 m North. Calculate the man's resultant displacement.

  1. As before, the rough sketch looks as follows:

  2. Note that the triangle formed by his separate displacement vectors and his resultant displacement vector is a right-angle triangle. We can thus use the Theorem of Pythagoras to determine the length of the resultant. Let x R represent the length of the resultant vector. Then:

    x R 2 = ( 40 m ) 2 + ( 30 m ) 2 x R 2 = 2 500 m 2 x R = 50 m
  3. Now we have the length of the resultant displacement vector but not yet its direction. To determine its direction we calculate the angle α between the resultant displacement vector and East, by using simple trigonometry:

    tan α = opposite side adjacent side tan α = 30 40 α = tan - 1 ( 0 , 75 ) α = 36 , 9
  4. The resultant displacement is then 50 m at 36,9 North of East.

    This is exactly the same answer we arrived at after drawing a scale diagram!

In the previous example we were able to use simple trigonometry to calculate the resultant displacement. This was possible since thedirections of motion were perpendicular (north and east). Algebraic techniques, however, are not limited to cases where the vectors to be combined are along the same straight line or at right angles to oneanother. The following example illustrates this.

A man walks from point A to point B which is 12 km away on a bearing of 45 . From point B the man walks a further 8 km east to point C. Calculate the resultant displacement.

  1. B A ^ F = 45 since the man walks initially on a bearing of 45 . Then, A B ^ G = B A ^ F = 45 (parallel lines, alternate angles). Both of these angles are included in the rough sketch.

  2. The resultant is the vector AC. Since we know both the lengths of AB and BC and the included angle A B ^ C , we can use the cosine rule:

    A C 2 = A B 2 + B C 2 - 2 · A B · B C cos ( A B ^ C ) = ( 12 ) 2 + ( 8 ) 2 - 2 · ( 12 ) ( 8 ) cos ( 135 ) = 343 , 8 A C = 18 , 5 km
  3. Next we use the sine rule to determine the angle θ :

    sin θ 8 = sin 135 18 , 5 sin θ = 8 × sin 135 18 , 5 θ = sin - 1 ( 0 , 3058 ) θ = 17 , 8

    To find F A ^ C , we add 45 . Thus, F A ^ C = 62 , 8 .

  4. The resultant displacement is therefore 18,5 km on a bearing of 062,8 .

More resultant vectors

  1. A frog is trying to cross a river. It swims at 3 m · s - 1 in a northerly direction towards the opposite bank. The water is flowing in a westerly direction at 5 m · s - 1 . Find the frog's resultant velocity by using appropriate calculations. Include a rough sketch of the situation in your answer.
  2. Sandra walks to the shop by walking 500 m Northwest and then 400 m N 30 E. Determine her resultant displacement by doing appropriate calculations.

Questions & Answers

how can chip be made from sand
Eke Reply
are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Richard is sitting on his chair and reading a newspaper three (3) meters away from the door
Jeo Reply
The fundamental frequency of a sonometer wire streached by a load of relative density 's'are n¹ and n² when the load is in air and completly immersed in water respectively then the lation n²/na is
Mukesh Reply
Properties of longitudinal waves
Sharoon Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now

Source:  OpenStax, Siyavula textbooks: grade 10 physical science [caps]. OpenStax CNX. Sep 30, 2011 Download for free at http://cnx.org/content/col11305/1.7
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Siyavula textbooks: grade 10 physical science [caps]' conversation and receive update notifications?