Vectors in two dimensions


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Vectors in two dimensions

  1. 1. 1 1. Vectors Physics Grade 11Everything Science
  2. 2. 2Vectors - revision A vector is a physical quantity with magnitude and direction.Vectors can be used to represent many physical quantities that have a magnitude anddirection, like forces.Vectors may be represented as arrows where the length of the arrow indicates themagnitude and the arrowhead indicates the direction of the vector. Everything Science
  3. 3. 3Vectors on the Cartesian planeWe can represent vectors on the Cartesian plane. The vectors can be placed anywhereon the Cartesian plane as long as the magnitude and direction of the vector is correctlyindicated. Everything Science
  4. 4. 4Vectors on the Cartesian plane continuedVectors can also be drawn at an angle to one of the axes. When we do this wespecify the angle as acting anti-clockwise from the positive x-axis. Everything Science
  5. 5. 5The resultant vectorThe resultant of a number of vectors is the single vector whose effect is the same as theindividual vectors acting together. Everything Science
  6. 6. 6Sketching the resultantWe can sketch vectors in two dimensions using the tail-to-head method. In this method thetail of one vector is placed at the head of the other vector. Everything Science
  7. 7. 7Sketching the resultant continuedWe can also first find the resultant in the x-direction and then find the resultant in they-direction before finding the final resultant. Everything Science
  8. 8. 8Tail-to-tail method of sketching the resultantWe can use the tail-to-tail method to sketch the resultant. This is illustrated below. Everything Science
  9. 9. 9Closed vector diagramsA closed vector diagram is a set of vectors drawn on the Cartesian using the tail-to-headmethod and that has a resultant with a magnitude of zero. This means that if the first vectorstarts at the origin the last vector drawn must end at the origin. The vectors form a closedpolygon, no matter how many of them are drawn. Here are a few examples of closed vectordiagrams: Everything Science
  10. 10. 10Finding the resultantUsing the algebraic techniques of vectoraddition from grade 10 and Pythagorastheorem we can find the magnitude of theresultant of vectors in two dimensions.We first find the resultant in the x-directionand then find the resultant in the y-direction.Finally we note that when we draw thesetwo vectors head-to-tail we get a rightangled triangle that has the resultant of thevectors as the hypotenuse.The magnitude of the resultant can becalculated algebraically or measuredgraphically from a scale diagram.The direction can be measured from a scalediagram or calculated using trigonometry. Everything Science
  11. 11. 11Components of vectorsWe can resolve any vector into components. This can be done algebraically orgraphically. We can use: and R x =R cos  R y =R sin to calculate the magnitude of the x- and y-components. Everything Science
  12. 12. 12Components of vectors continuedWe can extend this to vectors in two dimensions. Everything Science
  13. 13. 13For more practice see: ESBKF Everything Science