By W. von Oertzen
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Extra info for Detectors in Heavy-Ion Reactions
9: A vector can be broken into Manipulating vectors by manipulating components Because a vector can be represented by its components (once given a coordinate system) we should be able to relate our geometric understanding of vectors to their components. In practice, when push comes to shove, most calculations with vectors are done with components. a sum of vectors, each parallel to the axis of a coordinate system. , F x = Fx ıˆ . vectproject) 14 CHAPTER 2. ’ Similarly, [A − B]x yz = [(A x − Bx ), (A y − B y ), (A z − Bz )] Multiplying a vector by a scalar using components The vector A can be decomposed into the sum of three orthogonal vectors.
6 Scalar times a vector: Two forces acting on a particle are F1 = 100 Nˆı − 20 Nˆ and F2 = 40 Nˆ. If F1 is doubled, does the net force double? Solution Fnet F1 + F2 = (100 Nˆı − 20 Nˆ) + (40 Nˆ) = 100 Nˆı + 20 Nˆ = After F1 is doubled, the new net force F(net)2 is F(net)2 = = = 2F1 + F2 = 2(100 Nˆı − 20 Nˆ) + (40 Nˆ) 200 Nˆı − 40 Nˆ + 40 Nˆ 200 Nˆı = 2 (100 Nˆı + 20 Nˆ) F net No, the net force does not double. 1. 7 Magnitude and direction of a vector: The velocity of a car is given by v = (30ˆı + 40ˆ) mph.
For example, ıˆ × ˆ = k, Component form for the 2D cross product Just like the dot product, the cross product can be expressed using components. As can be verified by writing A = A x ıˆ + A y ˆ, and B = Bx ıˆ + B y ˆ and using the distributive rules: ˆ A × B = (A x B y − Bx A y )k. 7) Some people remember this formula by putting the components of A and B into a Ax A y matrix and calculating the determinant . , [A]x1 x2 = [A1 , A2 ]), the cross product is A × B = (A1 B2 − B2 A1 )eˆ3 . ” Example: Given that A = 1ˆı + 2ˆ and B = 10ˆı + 20ˆ then A × B = (1 · 20 − 2 · 10)kˆ = 0kˆ = 0.
Detectors in Heavy-Ion Reactions by W. von Oertzen