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Coordinate Systems

Coordinate Systems. Specifying position in 2D requires 2 numbers: Can use ( x , y ) Point P: (-5,3) Point Q: (-3,4) Use ( x,y,z ) in 3D Cartesian Coordinates Or, can use ( r, q ) Polar Coordinates. Coordinate Systems. Converting from Cartesian to Radial r 2 = x 2 + y 2

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Coordinate Systems

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  1. Coordinate Systems • Specifying position in 2D requires 2 numbers: • Can use (x,y) • Point P: (-5,3) • Point Q: (-3,4) • Use (x,y,z) in 3D • Cartesian Coordinates • Or, can use (r,q) • Polar Coordinates

  2. Coordinate Systems • Converting from Cartesian to Radial • r2 = x2+y2 • tan  = y/x • Radial to Cartesian • x = r cos  • y = r sin 

  3. Vectors A • A vector is a quantity that has both • Magnitude (size)and • Direction • Represented by an arrow • Length of the arrow is the magnitude • Angle of the arrow indicates direction • Symbol for a vector is a letter with an arrow over it (or a bold faced letter: A)

  4. Vectors • Two ways to specify a vector • Magnitude & Direction • Magnitude A • Direction , f • X & Y Components • Ax • Ay • Az

  5. Vectors • Can switch back and forth • Ax = A cos q • Ay = A sin q • A = (Ax2+Ay2)1/2 • tan  = Ay/Ax

  6. Vectors • Note that the position of the vector is not specified • All these vectors are equal

  7. Scalars • A scalar is just a number with no direction • Examples of scalars • Temperature • Mass • A person’s age • Distance • Speed

  8. Vector Addition • What does it mean to say R = A + B • Geometrical Answer • Put tail of B on head of A • R connects tail of A to head of B • Algebraic Answer…

  9. Vector Addition • Note that vector addition is commutative

  10. Vector Addition • Note that vector addition is commutative • …and associative

  11. Scalar Multiplication • Can change magnitude of vector by multiplication by scalar • Doesn’t change direction, except… • Negative scalar flips direction

  12. Unit Vectors • A Unit Vector has a magnitude of one • Will use ^ sign for unit vectors • We will define three special unit vectors i Unit vector pointing in x direction j Unit vector pointing in y direction k Unit vector pointing in z direction   

  13. Vector Components • Use unit vectors to break vectors down into components • Any vector is the sum of its components (which are also vectors)

  14. Vector Addition • Algebraic Answer (finally)

  15. Vector Subtraction • How do we subtract vectors? • Method 1 • Put tails of A and B together • C goes from head of B to head of C • Method 2 • Add –B to A • Algebraically • Just like addition

  16. Preview • Can multiply vectors in two ways • One gives a scalar • The other gives another vector

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