**VECTOR ADDITION FORMULA ON A SPHERICAL
SURFACE**
Md. Shah Alam Department of Physics Shahjalal
University of Science and Technology Sylhet, Bangladesh salam@sust.edu
**ABSTRACT** Vector addition formula on a plane
Euclidean surface is well known. In this paper we propose a vector
addition formula on a spherical surface. Motion on a sphere (for example,
the motion of an aircraft along the earth’s surface) can be studied using
this formula.
KEYWORDS: vector addition formula, spherical
surface.
**INTRODUCTION** We know that the surface of the
earth is spherical. A line on the earth’s surface reduces to a straight
line when its length is very small compared to the radius of the earth,
but its curvature is manifest when its length is large. Therefore, if we
draw a triangle on the earth’s surface it will be a plane triangle when
its sides are small and it will be a spherical triangle when its sides are
large. Our known vector addition formula is applicable on the plane
triangle, but not on the spherical triangle. In this paper we define
vectors on a spherical surface and their addition formula which can be
applied on the spherical triangle.
**DEFINITION OF VECTORS ON A
SPHERICAL SURFACE** Let us consider two points **A** and **B**
on the surface of a sphere. 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 the vector **BA** is from **B** to **A** along
the surface of the sphere. The magnitudes of these two vectors are same,
that is, the tangent of the angle subtended at the center of the sphere by
the curve **AB**.
**GENERATION OF VECTOR ADDITION FORMULA ON A
SPHERICAL SURFACE** Mushfiq Ahmad in his study of Lorentz
Transformation has defined a formula of addition of
vectors:^{1}
**A** **B** = (**A** + **B** + i**A**x**B**/c)
/ (1+**A.B**/c^{2}) .
Here represents new addition:
Let c = ic' and c' = 1 then
**A** **B** = (**A** + **B** + **A**x**B**) /
(1 -
**A.B**) . (1)
This is the vector addition formula on the spherical
surface.
Where **A**, **B** etc. are the tangents of the
angles subtended at the center of the sphere. From this formula we
can explained the following physical cases.
(i) When the angle between two vectors on the spherical
surface is 0° then their addition gives a curved line on the
spherical surface.
When angle between vectors **A **and **B** is 0° then
using equation (1) we get
**A** **B **= (**A** + **B**) / (1 - **A.B**) =
**D** (say)
or
D = (tanA + tanB) / (1-tanA tanB) = tan(A + B) .
(ii) When angle between the vectors **A** and **B** is
0° and angle between the vectors **B** and
**D** (**A****B** = **D**) is
180° then it represents a circle:
For this case using equation (1) we can write
**A** = **D** (-**B**)
Here, D = tan180°, B = tan90°, and
**A** = (**D** - **B** - **D**x**B**) / (1 +
**D.B**)
or
**A**^{2} = (**D**^{2} - 2**D.B**
+**B**^{2} - **D**^{2}**B**^{2} +
(**D.B**)^{2}) **/** (1 + 2**D.B** +
(**D.B**)^{2})
=(tan^{2}180°+2tan180°tan90°+tan^{2}90°-tan^{2}180°tan^{2}90°+(tan180°tan90°
)^{2})/(1-2tan180°tan90°)
= tan^{2} 90°
**A** = tan 90° .
Which is exactly same as shown in the fig.1.
(iii) When the sides of a equilateral spherical triangle are
very small then it behaves like a plane triangle.
Using equation (1) we get
Neglecting higher powers of D we get,
*or*, **cos** *d* = ½ = **cos**
60°
d = 60°
Which is the requirement of equilateral plane triangle.
The proposed vector addition formula is applicable on the
spherical triangle is shown below.
**EXAMPLES**
(i) Let us consider a sphere as shown in
fig.(2).
**A, B** and **D** are three vectors on the
surface of the sphere which form a spherical triangle and a, b, and d are
angles of this spherical triangle. In fig. 1 A = B = tan45°, d =
90° we want to find D = ?.
Here d = 90°
**A.B** = AB cos(180°-d) = - ABcos d = - cos 90°
Using eqn.(1) we get
**D** = (**A** + **B** + **A**x**B**) /
(1-**A.B**)
D^{2} = A^{2 }+ B^{2 }+ 2**A.B
**+ A^{2}B^{2} - (**A.B**)^{2 }/ (1 - 2A.B +
(**A.B**)^{2})
Using traditional method, we get, D = 60° where A, B, D
etc. are angles subtended at the center of the sphere.^{2}
(ii) Let, A = B = tan 45°, d = 60°
**A.B** = AB cos ( 180° - d ) = - cos d = - cos 60° =
-(1/2 )
(iii) Let, A = B = tan30° and d = 90°, D = ?
**A.B
**= AB cos (180°-d) = - cos d = - cos 90° = 0
**ADVANTAGES** Using this formula we can
explain rotation on a sphere clearly. In this representation A,
B etc. are tangents of the angles subtended at the center.
Traditionally, one represents them by the angles themselves subtended
at the center. Our method provides a direct vector algebra which
traditional method does
not.^{3}
**CONCLUSION** We have been able to
obtain a generalized formula of addition of vectors which in
one limit (when radius of the sphere ) reduces to addition
of vectors on a plane (Euclidean) surface, and in another limit, AxB
= 0 reduces to addition of tangents along a circle.
ACKNOWLEDGEMENT I am grateful to Mushfiq Ahmad, Dept.
of Physics, Rajshahi University, Bangladesh and Prof. Habibul Ahsan
of the Deparment of Physics, Shahjalal University of Science
and Technology, Sylhet, Bangladesh for their help and
advice.
REFERENCES [1] Ahmad, Mushfiq (Dept. of Physics, R.U.
Bangladesh). Private Communication. n.d. [2] Green, Robin M. Green.
Spherical Astronomy. Conn.: Cambridge UP, 1985. [3] Punmia, Dr.
B. C. Surveying. 9th ed. Vol. 3. New Delhi: Laxmi Publications (P)
Ltd.,1995.
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