Class 7: Vector and scalar, components Vector operations in - PowerPoint PPT Presentation

Class 7: Vector and scalar, components

Vector operations in components Multiplying a vector with a scalar: y x ‐ component y ‐ component A+ B A A x A y mA mA x mA y Addition of vectors: x ‐ component y ‐ component A A x A y B B x B y A A - B A+B A x +B x A y +B y Subtraction of vectors: B x ‐ component y ‐ component A A x A y B B x B y A ‐ B A x ‐ B x A y ‐ B y Vector × Vector  Complicated. No division between vectors

Two common ways to define a vector There are many ways to define a vector, the two most common ones are: By components or unit vectors By magnitude and angle (direction) y y  Is usually measured from the +x axis, but A y not necessary. |A|  j x x i A x |A| is independent of the choice of the coordinate system. A = (|A| ,  ) A = (A x , A y ) or A = A x i + A y j

Right angled triangle A c b a C B       2 2 2 2 2 a b c (or sin cos 1) opposite side   sin hypotenuse adjacent side   cos hypotenuse If you know two sides, or one side and  one angle, you know everything about opposite side sin    tan the right angled triangle.  adjacent side cos     o A B 90

Conversion between components and magnitude and direction an example y 3m By magnitude and direction: j   |v| = 5m/s  = 36.87 o x or i -4m |v| = 5m/s  = 180 o ‐ 36.87 o = 143.13 o By components or unit vectors: v x = ‐ 4m/s r y = 3m/s or v = ( ‐ 4 i + 3 j ) m/s

Same vector in different coordinate axes y y' 3m x' 5m ? j' i' ? = 90 o – 36.78 o – 20 o = 33.22 o 36.87 o 20 o x -4m r x' = 5 sin 33.22 o = 2.739m r y' = - 5 cos 33.22 o = - 4.183m manually! r = (2.739 i’ - 4.183 j’) m r x = -4m r y = 3m Magnitude of a vector is the same in all coordinate system (invariant).

Vector operations without a coordinate system Multiplying a vector with a scalar 3A A ‐ A Vector × Vector  Complicated. No division between vectors

Vector operations without a coordinate system Addition and subtraction of vectors Addition: Subtraction: A B A B ‐ B