<< Chapter < Page Chapter >> Page >
By the end of this section, you will be able to:
  • Apply analytical methods of vector algebra to find resultant vectors and to solve vector equations for unknown vectors.
  • Interpret physical situations in terms of vector expressions.

Vectors can be added together and multiplied by scalars. Vector addition is associative ( [link] ) and commutative ( [link] ), and vector multiplication by a sum of scalars is distributive ( [link] ). Also, scalar multiplication by a sum of vectors is distributive:

α ( A + B ) = α A + α B .

In this equation, α is any number (a scalar). For example, a vector antiparallel to vector A = A x i ^ + A y j ^ + A z k ^ can be expressed simply by multiplying A by the scalar α = −1 :

A = A x i ^ A y j ^ A z k ^ .

Direction of motion

In a Cartesian coordinate system where i ^ denotes geographic east, j ^ denotes geographic north, and k ^ denotes altitude above sea level, a military convoy advances its position through unknown territory with velocity v = ( 4.0 i ^ + 3.0 j ^ + 0.1 k ^ ) km / h . If the convoy had to retreat, in what geographic direction would it be moving?

Solution

The velocity vector has the third component v z = ( + 0.1 km / h ) k ^ , which says the convoy is climbing at a rate of 100 m/h through mountainous terrain. At the same time, its velocity is 4.0 km/h to the east and 3.0 km/h to the north, so it moves on the ground in direction tan −1 ( 3 / 4 ) 37 ° north of east. If the convoy had to retreat, its new velocity vector u would have to be antiparallel to v and be in the form u = α v , where α is a positive number. Thus, the velocity of the retreat would be u = α ( −4.0 i ^ 3.0 j ^ 0.1 k ^ ) km / h . The negative sign of the third component indicates the convoy would be descending. The direction angle of the retreat velocity is tan −1 ( −3 α / 4 α ) 37 ° south of west. Therefore, the convoy would be moving on the ground in direction 37 ° south of west while descending on its way back.

Got questions? Get instant answers now!

The generalization of the number zero to vector algebra is called the null vector    , denoted by 0 . All components of the null vector are zero, 0 = 0 i ^ + 0 j ^ + 0 k ^ , so the null vector has no length and no direction.

Two vectors A and B are equal vectors    if and only if their difference is the null vector:

0 = A B = ( A x i ^ + A y j ^ + A z k ^ ) ( B x i ^ + B y j ^ + B z k ^ ) = ( A x B x ) i ^ + ( A y B y ) j ^ + ( A z B z ) k ^ .

This vector equation means we must have simultaneously A x B x = 0 , A y B y = 0 , and A z B z = 0 . Hence, we can write A = B if and only if the corresponding components of vectors A and B are equal:

A = B { A x = B x A y = B y A z = B z .

Two vectors are equal when their corresponding scalar components are equal.

Resolving vectors into their scalar components (i.e., finding their scalar components) and expressing them analytically in vector component form (given by [link] ) allows us to use vector algebra to find sums or differences of many vectors analytically (i.e., without using graphical methods). For example, to find the resultant of two vectors A and B , we simply add them component by component, as follows:

R = A + B = ( A x i ^ + A y j ^ + A z k ^ ) + ( B x i ^ + B y j ^ + B z k ^ ) = ( A x + B x ) i ^ + ( A y + B y ) j ^ + ( A z + B z ) k ^ .

In this way, using [link] , scalar components of the resultant vector R = R x i ^ + R y j ^ + R z k ^ are the sums of corresponding scalar components of vectors A and B :

Questions & Answers

I'm interested in biological psychology and cognitive psychology
Tanya Reply
what does preconceived mean
sammie Reply
physiological Psychology
Nwosu Reply
How can I develope my cognitive domain
Amanyire Reply
why is communication effective
Dakolo Reply
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.
miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
Wekolamo Reply
please i need answer
Wekolamo
because it helps many people around the world to understand how to interact with other people and understand them well, for example at work (job).
Manix Reply
Agreed 👍 There are many parts of our brains and behaviors, we really need to get to know. Blessings for everyone and happy Sunday!
ARC
A child is a member of community not society elucidate ?
JESSY Reply
Isn't practices worldwide, be it psychology, be it science. isn't much just a false belief of control over something the mind cannot truly comprehend?
Simon Reply
compare and contrast skinner's perspective on personality development on freud
namakula Reply
Skinner skipped the whole unconscious phenomenon and rather emphasized on classical conditioning
war
explain how nature and nurture affect the development and later the productivity of an individual.
Amesalu Reply
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills
Zyryn Reply
good👍
Jonathan
and having a good philosophy of the world is like a sandwich and a peanut butter 👍
Jonathan
generally amnesi how long yrs memory loss
Kelu Reply
interpersonal relationships
Abdulfatai Reply
What would be the best educational aid(s) for gifted kids/savants?
Heidi Reply
treat them normal, if they want help then give them. that will make everyone happy
Saurabh
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 2

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'University physics volume 1' conversation and receive update notifications?

Ask