Review of Vectors

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Definitions:

  • Scalar: Any quantity possessing magnitude (size) only, such as mass, volume, temperature
  • Vector: Any quantity possessing both magnitude and direction, such as force, velocity, momentum

Vector Addition:

Vector addition may be done several ways including, Graphical Method, Trigonometric Method, and Component Method. We will be reviewing only the Component Method, as that is the method which will be used in the course. Other methods are detailed in your textbook.

Vector Addition – Component Method: (2-dimensional)

  1. The component method will follow the procedure shown below:Choose an origin, sketch a coordinate system, and draw the vectors to be added (or summed).
  2. Break (resolve) each vector into it’s “x” and “y” components, using the following relationships: Ax = A cosine θ, and,Ay = A sine θ, where A is the vector, and θ is the vector’s angle with respect to the positive x-axis.  If angle other than that with positive x-axis is given, use appropriate sine or cosine relationship.
  3. Sum all the x-components and all the y-components obtaining a net resultant  Rx, and Ry vectors. Rx = Ax + Bx + Cx + . . .,&, Ry = Ay + By + Cy + …
  4. Recombine Rx and Ry to obtain the final resultant vector (magnitude and direction) using

vectimg1

 

 

See Example below


 

Example – Vector Additionvectimg2

Three ropes are tied to a small metal ring. At the end of each rope three students are pulling, each trying to move the ring in their direction. If we look down from above the students, the forces and directions they are applying the forces are as follows: (See diagram to the right)

Find the net (resultant) force (magnitude and direction) on the ring due to the three applied forces.

Choose origin, sketch coordinate system and vectors (done above)
Resolve vectors into x & y components (See Diagram)
Ax = 30 lb cos 37o = + 24.0 lbs ; Ay = 30 lb sin 37o = + 18.1 lb
Bx = 50 lb cos135o = – 35.4 lbs ; By = 50 lb sin135o = + 35.4 lb
Cx = 80 lb cos240o = – 40.0 lbs ; Cy = 80 lb sin240o = – 69.3 lb

vectimg3

Sum x & y components to find resultant Rx and Ry forces.
Rx = 24.0 lbs – 35.4 lbs – 40.0 lbs = -51.4 lbs
Ry =18.1 lbs + 35.4 lbs – 69.3 lbs = -15.8 lbs

vectimg4

‘Recombine’ (add) Rx and Ry to determine final resultant vector.

Thus the resultant force on the ring is 53.8 pounds acting at an angle of 197.1 degrees.

Vector review problems