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notes - 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 i + A i (commutative law) 2. A i + ( B i + C i ) = ( A i + B i ) + C i (associative law) 3. ϕ A = B is a vector 1
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1.2 Scalar product The scalar product is defined to be a scalar quantity obtained by multiplying the magnitudes of two vectors: S = A · B = AB cos θ = i A i B i (9) where θ is the angle between the vectors. The magnitude of a vector can be calculated | R | = R · R = i R 2 i (10) The scalar product has certain useful properties: 1. A · B = B · A (commutative). 2. A · B = 0 if A or B = 0, or θ = 90 degrees (the vectors are orthogonal ). 3. A · ( B + C ) = A · B + A · C (distributive) 4. ( ϕ A ) · B = ϕ ( A · B ) (associative with scalar multiplication) example: Two position vectors are A = ˆ i + 2 ˆ j - 2 ˆ k and B = 4 ˆ i + 2 ˆ j - 3 ˆ k . Find the magnitude of the vector from point A to B, the angle θ between A and B , and the component of B in the direction of A solution: The vector from point A to B is B - A . By components: B - A = (4 - 1) ˆ i + (2 - 2) ˆ j + ( - 2 + 3) ˆ k = 3 ˆ i - ˆ k (11) The magnitude of this vector is | B - A | = 9 + 1 = 10 (12) From the scalar product Eq. 9: cos θ = A · B AB = 4 + 4 + 6 9 29 = 0 . 8666 (13) The component of B in the direction of A is B cos θ , so B cos θ = A · B A = 14 3 = 4 . 67 (14) 2
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1.3 Vector product The vector (or cross) product is defined to be a vector quantity: C = A × B (15) i.e. C has a magnitude and direction. The magnitude of C is C = AB sin θ (16) (for 0 < θ < π ), and it’s direction is determined by the right-hand rule, defined along the normal direction ˆ n . Thus, for example, one can define the area of a parallelogram with two sizes B and C as area A = B × C = ( BC sin θ n (17) The components of C are defined by the relation C i j,k ε ijk A j B k (18) where ε ijk is the Levi-Civita symbol, defined: ε 123 = ε 231 = ε 312 = 1 (19) ε 132 = ε 213 = ε 321 = - 1 ε ijk = 0 otherwise . Alternatively, the vector product can be represented by a determinate: C = ˆ i ˆ j ˆ k A x A y A z B x B y B z (20) in other words C x = A y B z - A z B y (21) C y = A z B x - A x B z C z = A x B y - A y B x The vector product has certain useful properties: 1. A × B = - B × A (anticommutative).
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