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- 1. Vectors<br />
- 2. Surveyors use accurate measures of magnitudes and directions to create scaled maps of large regions.<br />Vectors<br />
- 3. Identifying Direction<br />N<br />50o<br />60o<br />E<br />W<br />60o<br />60o<br />S<br />A common way of identifying direction is by reference to East, North, West, and South. (Locate points below.)<br />Length = 40 m<br />40 m, 50o N of E<br />40 m, 60o N of W<br />40 m, 60o W of S<br />40 m, 60o S of E<br />
- 4. Identifying Direction<br />N<br />45o<br />N<br />E<br />W<br />E<br />W<br />50o<br />S<br />S<br />Write the angles shown below by using references to east, south, west, north.<br />500 S of E<br />450 W of N<br />
- 5. 90o<br />90o<br />R<br />q<br />50o<br />180o<br />180o<br />0o<br />0o<br />270o<br />270o<br />Vectors and Polar Coordinates<br />Polar coordinates (R,q) are an excellent way to express vectors. Consider the vector 40 m, 500 N of E, for example.<br />40 m<br />R is the magnitude and q is the direction.<br />
- 6. 90o<br />0o<br />180o<br />50o<br />60o<br />60o<br />60o<br />3000<br />210o<br />270o<br />120o<br />Vectors and Polar Coordinates<br />Polar coordinates (R,q) are given for each of four possible quadrants:<br /> = 40 m, 50o<br />= 40 m, 120o<br />= 40 m, 210o<br />= 40 m, 300o<br />
- 7. y<br />(-2, +3)<br />(+3, +2)<br />+<br />+<br />x<br />-<br />Right, up = (+,+)<br />Left, down = (-,-)<br /> (x,y) = (?, ?)<br />-<br />(+4, -3)<br />(-1, -3)<br />Rectangular Coordinates<br />Reference is made to x and y axes, with + and -numbers to indicate position in space.<br />
- 8. R<br />y<br />q<br />x<br />Trigonometry Review<br />Application of Trigonometry to Vectors <br />Trigonometry<br />y = R sin q<br />x = R cos q<br />R2 = x2 + y2<br />
- 9. q<br />Finding Components of Vectors<br />A component is the effect of a vector along other directions. The x and y components of the vector are illustrated below.<br />= A cosq<br /> = A sin q<br /> Finding components:<br />Polar to Rectangular Conversions<br />
- 10. Example 2:A person walks 400.0 m in a direction of 30.0oSof W (210o). How far is the displacement westand how far south?<br />30o<br />400 m<br />The x-component (W) is adjacent:<br /> = -A cosq<br />The y-component (S) is opposite:<br />= -A sinq<br />
- 11. Vector Addition<br />Resultant ( ) <br /> - sum of two or more vectors.<br />Vector Resolutions: <br />01. GRAPHICAL SOLUTION <br /> – use ruler and protractor to draw and measure the scaled magnitude and angle (direction), respectively.<br /> 02. ANALYTICAL SOLUTION<br /> - use trigonometry<br />
- 12. q<br />Example 11:A bike travels 20 m, E then 40 m at 60o N of W, and finally 30 m at 210o. What is the resultant displacement graphically?<br />Graphically, we use ruler and protractor to draw components, then measure the Resultant R,q<br />C = 30 m<br />B = 40 m<br />30o<br />R<br />60o<br />f<br />A = 20 m, E<br />R = (32.6 m, 143.0o)<br />Let 1 cm = 10 m<br />
- 13. Cy<br />30o<br />R<br />Ry<br />60o<br />f<br />0<br />q<br />Ax<br />Rx<br />Bx<br />Cx<br />A Graphical Understanding of the Components and of the Resultant is given below:<br />Note: Rx = Ax + Bx + Cx<br />By<br />B<br />Ry = Ay + By + Cy<br />C<br />A<br />
- 14. Resultant of Perpendicular Vectors<br />Finding resultant of two perpendicular vectors is like changing from rectangular to polar coord.<br />R<br />y<br />q<br />x<br />R is always positive; q is from + x axis<br />
- 15. First Consider A + B Graphically:<br />B<br />B<br />R<br />B<br />A<br />A<br />Vector Difference<br />For vectors, signs are indicators of direction. Thus, when a vector is subtracted, the sign (direction) must be changed before adding.<br />R = A + B<br />
- 16. Now A – B: First change sign (direction) of B, then add the negative vector.<br />B<br />B<br />-B<br />A<br />R’<br />-B<br />A<br />A<br />Vector Difference<br />For vectors, signs are indicators of direction. Thus, when a vector is subtracted, the sign (direction) must be changed before adding.<br />
- 17. Subtraction results in a significant difference both in the magnitude and the direction of the resultant vector. |(A – B)| = |A| - |B|<br />Comparison of addition and subtraction of B<br />B<br />B<br />A<br />R<br />R’<br />-B<br />B<br />A<br />A<br />Addition and Subtraction<br />R = A + B<br />R’ = A - B<br />
- 18. A – B; B - A<br />+A<br />-A<br />+B<br />-B<br />A 2.43 N<br />B 7.74 N<br />Example 13.Given A = 2.4 km, N and B = 7.8 km, N: find A – B and B – A.<br />A - B<br />B - A<br />R<br />R<br />(2.43 N – 7.74 S)<br />(7.74 N – 2.43 S)<br />5.31 km, S<br />5.31 km, N<br />

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