Journal of Theoretics Vol.3-4 ALGEBRA OF VECTORS ON A SPHERICAL SURFACE Md. Shah Alam Deptartment of Physics Shahjalal University of Science and Technology Sylhet, Bangladesh [email protected] ABSTRACT The algebra of vectors on a plane Euclidean surface is well known. In this paper we represented the conception of vectors on a spherical surface which is different from the vectors on a plane Euclidean surface and we studied the necessary algebra of the vectors on a spherical surface. KEYWORDS: Vectors on a spherical surface, Euclidean surface. INTRODUCTION Let us consider two points on the surface of a sphere whose radius is unit. We draw a great circle on the surface of the sphere through the points A and B. AB and BA are two vectors on the spherical surface. The direction of AB vector is from A to B along the surface of the sphere and the direction of BA vector is from B to A along the surface of the sphere. The magnitudes of these vectors are same, which is the tangent of the angle subtended at the center of the sphere by the curve AB or BA. We called these vectors are spherical vectors. In this paper we studied the algebra of these spherical vectors. ALGEBRA OF SPHERICAL VECTORS (i) Equal vector: Two spherical vectors are equal or the same if they have the same angle subtended at the center of the sphere and same direction i.e. same rotation. (ii) Geometric Addition: If P and Q are two spherical vectors such that the vector P is from A to B and the vector Q is from B to C on the surface of a sphere shown in figure (1) then their addition (AB ⊕ BC = AC) can be written as [1] B P Q

A C

R Figure (1)

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P+Q+P×Q P⊕Q=

=R

(1)

1 – P.Q ⊕ → Represents the vector addition on a spherical surface We know that the geometric addition of vectors in the plane surface is obeyed the parallelogram law [2]. From equation (1) we can also say that the geometric addition of vectors on a spherical surface also obeyed the parallelogram law. (iii) Subtraction: Let P and Q are two spherical vectors as shown in Figure (2), then their subtraction can be written as: P ⊕ (-Q) + P × (-Q) P ⊕ (-Q) = 1 – P.(-Q) P-Q- P×Q or, P (-) Q = S =

(2) 1 + P.Q

(In the Figure P = AB, Q = BC, R = AC, -Q = BD, S = AD) Therefore from the figure (2) we can write: AB ⊕ BC = AC and AB ⊕ BD = AD BC = - BD ∴AB (-) BC = AD or, P (-) Q = S B Q

-Q P

C

D R

A

S

Figure (2) (iv) Scalar and Vector Products: If P and Q are two spherical vectors then their scalar product is:

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P.Q = PQ cosine θ = (r tan p)(r tan q) cosine θ Where: r → Radius of the sphere p → Angle subtended at the center of the sphere by the vector P q → Angle subtended at the center of the sphere by the vector Q θ → Angle between the vectors P and Q on the spherical surface that is called spherical angle [3]. If r = 1, then: P.Q = (tan p)(tan q) cosine θ The vector product of P and Q is: P × Q = PQ sin θ = (r tan p)(r tan q) sin θ If r = 1, then: P × Q = (tan p)(tan q) sin θ (v) Unit spherical vector: The magnitude of a spherical vector P is: P = r tan p Where: r → Radius of the sphere p → Angle subtended at the center of the sphere by the vector P If r = 1, then P = tan p, if p = 1° Then this P we called unit spherical vector. (vi) Components: Let us consider two axes along the surface of a sphere one is horizontal rotational axis and another is vertical rotational axis as shown in figure (3). Then any Spherical vector on the surface of the sphere can be expressed as:

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P=ah+bv Where h is the unit vector along the Horizontal Rotational axis and v is the unit vector along the vertical rotational axis. a and b are two scalar quantities. Vertical Rotational axis

Horizontal Rotational axis

Figure (3) (vii) Algebraic Addition: If P1 = p1h + q1v, P2 = p2h + q2v are two spherical vectors then according to equation number (1) their addition is: P1 ⊕ P2 =

P1 + P2 + P1× P2

or, P1 ⊕ P2 =

(p1h+q1v) + (p2h+q2v) + (p1h+q1v) × (p2h+q2v) =

1 – P1.P2

1 – (p1h+q1v).(p2h+q2v)

(p1+p2)h + (q1+q2)v +(p1q2+p2q1) 1 – (p1p2 +q1q2)

CONCLUSION We have represented the conception of vectors on spherical surface whose we call spherical vectors. The addition, subtraction, scalar and vector products, unit vectors and components of spherical vectors are studied which we call the algebra of spherical vectors. We have found a satisfactory algebra of spherical vectors. Therefore it can be used in Physics and Mathematics in relevant places.

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ACKNOWLEDGEMENT I am grateful to Mushfiq Ahmad, Dept. of Physics, University of Rajshahi, Rajshahi, Bangladesh and Prof. Habibul Ahsan, Dept. of Physics, Shahjalal University of Science and Technology, Sylhet for their help and advise. REFERENCES: [1] Md. Shah Alam, “Vector Addition Formula on a Spherical Surface,” Journal of Theoretics, Vol.3 No.3 June/July 2001. [2] George B. Thomas, Jr. and Ross L. Finney, Calculus and Analytic Geometry (AddisonWesley 1998). [3] Robin M. Green, Spherical Astronomy (Cambridge University press 1995).

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