class1 - 1 Review Vector Analysis REFERENCES Arya Chapter 5...

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1 Review: Vector Analysis REFERENCES: Arya Chapter 5. 1.1 Vectors A scalar has only a magnitude. A vector has both a magnitude and a direction. In this course, vectors will be represented in two ways (bold, or overarrow): R = -→ R, (1) or in the special case of a unit vector, with a hat (e.g. ˆ i ). The magnitude of a vector is represented R = | R | = | -→ R | (2) Vectors can be written in components. For example, in two dimensions: R = ( R x , R y ) (3) and R x = R cos( θ ) R y = R sin( θ ) (4) where R = R 2 x + R 2 y tan θ = R y R x (5) We also denote vectors with unit vectors : For example in 3D cartesian coordinates: ˆ i = (1 , 0 , 0) , ˆ j = (0 , 1 , 0) , ˆ k = (0 , 0 , 1) (6) where ˆ i is the unit vector associated with the x-axis direction, ˆ j the y-axis direction, and ˆ k the z-axis direction. Each has unit magnitute | ˆ i | = | ˆ j | = | ˆ k | = 1 (7) A general vector in 3D can then be represented as: R = ( R x , R y , R z ) = R x ˆ i + R y ˆ j + R z ˆ k (8) Adding and subtracting vectors can be done component-wise. From this, certain proper- ties follow. Consider vectors A , B and C and scalar ϕ : 1. A i + B i = B
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