P2F11-Week6-Barwick-ans

P2F11-Week6-Barwick-ans - P2 Week 6 Relative Velocity and...

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P2 Week 6 Relative Velocity and Dot Product Sections 3.5 and 1.10
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Review of Adding Vectors: Graphically and Algebraically l Draw the vectors “tip-to-tail” l The resultant is drawn from the origin of to the end of the last vector l Measure the length of and its angle l Use the scale factor to convert length to actual magnitude A ! R !
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Adding Vectors Using Unit Vectors l Using l Then l and so R x = A x + B x and R y = A y + B y ( ) ( ) ( ) ( ) ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ x y x y x x y y x y A A B B A B A B R R = + + + = + + + = + R i j i j R i j R i j ! ! ! 2 2 1 tan y x y x R R R R R ! " = + = = + R A B ! ! ! Careful with inverse trig functions! Several possible answers. Pay attention to signs of Ry and Rx
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Scalar (or Dot) Product of Two Vectors: Method 1 l The scalar product of two vectors is written as l It is also called the dot product l l θ is the angle between A and B ! A B ! ! A B cos ! " # A B ! ! The dot product is the multiplication of two quantities: (1) the projection of the vector B on vector A and (2) the magnitude of vector A. This is a very useful thing to know. There are many physical situations where it is important to know the fraction of one vector that lies in the direction of another.
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Scalar Product, cont l The scalar product produces a scalar, not a vector l The scalar product is commutative and obeys the distributive law of multiplication l l Since cos( θ ) symmetric about zero, it does not matter if you assign a positive or negative value to the angle l Theta, the angle between the vectors, should lie between 0 and 180 degrees, or 0 and -180 degrees l The dot product of perpendicular vectors is zero, independent of the magnitudes of the vectors. You can figure out if two vectors are perpendicular (or orthogonal) if the dot product is zero. ! A ! B = ! B ! A ! A ( ! B + ! C ) = ! A ! B + ! A ! C commutative distributive
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Dot Products of Unit Vectors l l Using component form with vectors: ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ! = ! = ! = ! = ! = ! = i i j j k k i j i k j k 1 0 ˆ ˆ ˆ ˆ ˆ ˆ = + + = + + = + + A i j k B i j k A B ! ! ! ! i x y z x y z x x y y z z A A A B B B A B A B A B Projection of unit vector on identical vector is 1 Projection of vector at right angle to another is 0 2nd method to compute the dot product of two vectors .
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l What is scalar product of (1,–1) and (2,3) l Since the information given is a component representation of vector, use l A x B x +A y B y = (1)(2)+(–1)(3)=2-3=–1 l The minus sign is correct, and indicates that projection of vector B is in the opposite direction of A. l What is dot product of vector with magnitude 4 pointing due east, and another with magnitude 2 pointing 60 deg north of west. l
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P2F11-Week6-Barwick-ans - P2 Week 6 Relative Velocity and...

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