<< Chapter < Page Chapter >> Page >
By the end of this section, you will be able to:
  • Describe vectors in two and three dimensions in terms of their components, using unit vectors along the axes.
  • Distinguish between the vector components of a vector and the scalar components of a vector.
  • Explain how the magnitude of a vector is defined in terms of the components of a vector.
  • Identify the direction angle of a vector in a plane.
  • Explain the connection between polar coordinates and Cartesian coordinates in a plane.

Vectors are usually described in terms of their components in a coordinate system . Even in everyday life we naturally invoke the concept of orthogonal projections in a rectangular coordinate system. For example, if you ask someone for directions to a particular location, you will more likely be told to go 40 km east and 30 km north than 50 km in the direction 37 ° north of east.

In a rectangular (Cartesian) xy -coordinate system in a plane, a point in a plane is described by a pair of coordinates ( x , y ). In a similar fashion, a vector A in a plane is described by a pair of its vector coordinates. The x -coordinate of vector A is called its x -component and the y -coordinate of vector A is called its y -component. The vector x -component is a vector denoted by A x . The vector y -component is a vector denoted by A y . In the Cartesian system, the x and y vector components    of a vector are the orthogonal projections of this vector onto the x - and y -axes, respectively. In this way, following the parallelogram rule for vector addition, each vector on a Cartesian plane can be expressed as the vector sum of its vector components:

A = A x + A y .

As illustrated in [link] , vector A is the diagonal of the rectangle where the x -component A x is the side parallel to the x -axis and the y -component A y is the side parallel to the y -axis. Vector component A x is orthogonal to vector component A y .

Vector A is shown in the x y coordinate system and extends from point b at A’s tail to point e and its head. Vector A points up and to the right. Unit vectors I hat and j hat are small vectors pointing in the x and y directions, respectively, and are at right angles to each other. The x component of vector A is a vector pointing horizontally from the point b to a point directly below point e at the tip of vector A. On the x axis, we see that the vector A sub x extends from x sub b to x sub e and is equal to magnitude A sub x times I hat. The magnitude A sub x equals x sub e minus x sub b. The y component of vector A is a vector pointing vertically from point b to a point directly to the left of point e at the tip of vector A. On the y axis, we see that the vector A sub y extends from y sub b to y sub e and is equal to magnitude A sub y times j hat. The magnitude A sub y equals y sub e minus y sub b.
Vector A in a plane in the Cartesian coordinate system is the vector sum of its vector x - and y -components. The x -vector component A x is the orthogonal projection of vector A onto the x -axis. The y -vector component A y is the orthogonal projection of vector A onto the y -axis. The numbers A x and A y that multiply the unit vectors are the scalar components of the vector.

It is customary to denote the positive direction on the x -axis by the unit vector i ^ and the positive direction on the y -axis by the unit vector j ^ . Unit vectors of the axes , i ^ and j ^ , define two orthogonal directions in the plane. As shown in [link] , the x - and y - components of a vector can now be written in terms of the unit vectors of the axes:

{ A x = A x i ^ A y = A y j ^ .

The vectors A x and A y defined by [link] are the vector components of vector A . The numbers A x and A y that define the vector components in [link] are the scalar component     s of vector A . Combining [link] with [link] , we obtain the component form of a vector :

A = A x i ^ + A y j ^ .

If we know the coordinates b ( x b , y b ) of the origin point of a vector (where b stands for “beginning”) and the coordinates e ( x e , y e ) of the end point of a vector (where e stands for “end”), we can obtain the scalar components of a vector simply by subtracting the origin point coordinates from the end point coordinates:

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 8

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