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Unformatted text preview: Chapter 3 Vectors Physics deals with many quantities that have both Size Direction VECTORS !!!!! y r (x,y) (r,) x E.g. Displacement, Velocity, Acceleration, Force, Torque x and y components of motion are independent-"LINEARITY" Componets v x = v cos v y = v sin ^ v = (v x , v y ) = v x i + v y ^ j Unit vectors Displacement Vector: now in 2-D
Displacement is same for all three vectors, ie same direction and same length Three different paths give the same displacement Definitions
SCALAR provides information about how large a measurement is Gives one item of information, magnitude, temperature=T VECTOR provides information about how large a measurement is and the direction of the measurement Ordered pair numbers required (x,y) (r,) Addition of displacement vectors Graphically s =a+b Adding more than two vectors graphically ADDITIONAL VECTOR PROPERTIES
A vector can be moved (in a diagram) so long as the magnitude and direction is unchanged Vectors may be expressed as ordered numbers, polar form or in unit vector form Vector subtraction may be accomplished by multiplying the subtracted vector by 1 and using the technique for adding Subtraction of Vectors -graphically d = a + (-b) whereas c = a + b d = a-b VECTOR ALGEBRA Vectors are added or subtracted according to the rules for ordered pairs (ax,ay)-(bx,by) coordinates!
See blackboard a = (a x ,a y ) = (4.2, - 1.5) a +b =? b = (bx ,by ) = (-1.6, 2.9) Vectors are added or subtracted according Ordered pair numbers (x,y) (r,) Vector Components cos = sin = tan = ax
2 2 ax + a y If you use your calculator to determine angle you will find tan-1 (-5/7)= -35o 325o ay
2 2 ax + a y ay ax Unit Vector Notation a = (4.2, - 1.5) or a = 4.2^ + -1.5^ i j b = (-1.6, 2.9) or b = -1.6^ + 2.9^ i j
Unit vectors have magnitude 1 and are "unitless" ... they only give the direction!!!! VECTOR NOTATION Components for a vector may be expressed in unit vector notation ^ i is a unit vector in the x direction ^ is a unit vector in the y direction j ^ k is a unit vector in the z direction Bold type or an arrow above the symbol denotes a vector; e.g., A or A The magnitude of the above vector is designated A VECTOR ALGEBRA cont. Vectors are added or subtracted according to the rules for ordered pairs (ax,ay)-(bx,by)-(cx,cy)--coordinates! a = 4.2^ + -1.5^j or (4.2, - 1.5) i b = -1.6^ + 2.9^j or ( -1.6, 2.9) i c = 0.0^ + -3.7^j or (0.0,-3.7) i
Rule for graphical addition is implied!!!! Ant Example
Find resultant Find the vector home Vector Multiplication Scalar multiplication b = 2a Vectors can be multiplied in two ways A dot product of two vectors results in a scalar c = a b A cross product of a vector results in another vector c = a b Vectors are NEVER divided! The Scalar or Dot Product Multiplication of two vectors resulting in a scalar A B = AB cos Example W = F d , for constant force Some Properties of the Dot Product
Dot products commute The square of a vector ^ ^ j j ^ ^ i i = ^ ^ = k k =1 Unit vector products ^ j j ^ ^ ^ i ^ = ^k = i k = 0 A B = B A EXAMPLE 2 What is the dot product of ^ A = 5.0 i + 3.0 ^ j B = (2.0, 155 ) The VECTOR PRODUCT or CROSS PRODUCT Vector multiplication yielding another vector Yields a vector which has a direction determined by the right hand rule Yields a vector perpendicular to the plane containing the other two vectors The cross product DOES NOT commute C = A B -C = B A = r F torque MAGNITUDE OF THE CROSS PRODUCT C = AB sin DIRECTION OF THE CROSS PRODUCT The right hand rule determines the direction of the cross product Unit Vector Cross Products Using the definition of cross product and righthand rule:
^ ^ ^ i^ j = k = - j i^ ^ ^ ^ j k = i^ = -k i^ ^ ^ ^ k i^ = j = -i^ k ^ i i^ = ^ ^ = k k = 0 j j ^ ^ EXAMPLE 3 Calculate the cross product of ^ A = 5.0 i^ + 3.0 j B = (2.0, 155 )
Write component form B in Projectile Motion Classic 2-D problem....Eqs. Of Motion? v sin 2 x R = g
2 0 ...
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