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An Introduction to the to the concepts of Vectors and Scalars

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- 1. Topic 1 – Physical Measurement 1.3 – Vectors and Scalars
- 2. Vectors and Scalars <ul><li>In Physics some quantities have values that depend on direction and some do not.
- 3. A quantity that has a direction associated with it is called a vector. </li><ul><li>Example velocity. </li></ul><li>A quantity that has no direction is called a scalar. </li><ul><li>Example mass </li></ul><li>For a vector quantity a “+” or “-” sign shows direction
- 4. For a scalar quantity a “-” sign is (usually) meaningless </li></ul>
- 5. Vectors and Scalars <ul><li>Sort the list of quantities below into a table of vectors and scalars. </li><ul><li>Mass Displacement
- 6. Time Energy
- 7. Velocity Speed
- 8. Distance Acceleration
- 9. Amount of Substance Force </li></ul></ul>
- 10. Representing Vectors <ul><li>A vector is generally represented in equations by using an underline, or by an over-arrow. </li><ul><li>Some textbooks use bold fonts to show vectors.
- 11. Example: </li></ul></ul>
- 12. Representing Vectors <ul><li>Vectors are represented in diagrams by an arrow (often drawn to a scale) pointing in the correct direction but attached by its tail to the point at which it acts. </li><ul><li>Example </li></ul></ul><ul>W=45N </ul><ul>F=20N </ul>
- 13. Combining Vectors <ul><li>Often in Physics we need to find what the total effect of a group of vectors is.
- 14. The one vector that can be used to cause the same effect as the group is called the resultant vector.
- 15. Often (but not always) the resultant is drawn as a double headed arrow. </li></ul>
- 16. Adding Vectors <ul><li>In order to add vectors graphically we need to draw them to scale following some basic rules. </li><ul><li>Decide on a scale and stick to it!
- 17. Draw one vector and label it if necessary.
- 18. Draw the next vector starting at the head of the first one.
- 19. Carry on doing this until all vectors are drawn.
- 20. Join the starting point to the end point and mark this as the resultant.
- 21. Measure the resultant’s length and convert this using your scale into the correct units.
- 22. Measure the direction of the resultant from a sensible datum. </li><ul><li>e.g. above horizontal, right of vertical, below the direction of vector 1. </li></ul></ul></ul>
- 23. Adding Vectors <ul>Example: <ul><li>What is the resultant of a 7N horizontal force to the right and a 12N force vertically down? </li></ul></ul><ul>Scale 1cm=1N </ul><ul>13.9cm = 13.9N </ul><ul>60 o </ul><ul>Answer: 13.9N @ 60 o below Horizontal </ul>
- 24. Subtracting Vectors <ul><li>Subtracting vectors follows exactly the same process as addition. </li><ul><li>Example: </li><ul><li>two vectors a and b are defined as a = 5ms -1 horizontal to the right b = 10ms -1 to the right.
- 25. What is r = a – b ? </li></ul></ul></ul>
- 26. Subtracting Vectors <ul><li>a – b = a + (- b ) </li><ul><li>- b is the same magnitude (size) as b but acts in the opposite direction to b . </li></ul></ul><ul>a </ul><ul>b </ul><ul>-b </ul><ul>a </ul><ul>-b </ul><ul>r </ul><ul>r = a + b </ul><ul>r = a - b </ul><ul>-b </ul><ul>r </ul>
- 27. Combining Vectors Mathematically <ul><li>Consider any vector a . </li><ul><li>It can be resolved (broken down) into 2 orthogonal components a x and a y .
- 28. This is done using the sine and cosine identies </li></ul></ul>a a x a y θ
- 29. Combining Vectors Mathematically <ul><li>The same process can also be used to resolve vector b into its components.
- 30. To find the resultant r is then simply a matter of summing the x components and the y components. </li><ul><li>r x = a x + b x + … r y = a y + b y + ... </li></ul><li>These resultant components can then be combined using Pythagoras and tangent to find the magnitude and direction. </li></ul>
- 31. Combining Vectors Mathematically <ul><li>When working with large numbers of vectors it is often useful to represent each vector as a column matrix.
- 32. This then does not require x and y subscripts and can be easier to follow. </li><ul><li>This is almost essential if working with x, y and z!
- 33. Example </li></ul></ul>

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