In spherical polar coordinates (l, q, a), q denotes the polar angle around z-axis and a denotes the azimuthal angle raised from x-axis . Then the y-component of P is given by

1. Basics of Cartesian Coordinate Systems (basic)

At its heart, the Cartesian Coordinate System is a framework used to uniquely identify the position of a point in space. Imagine the corner of a room where the floor meets two walls; this corner is the origin (0, 0, 0). From this origin, we draw three mutually perpendicular lines called the X, Y, and Z axes. This system allows us to transform a physical location into a set of numbers, which is essential for everything from tracking the movement of planets to understanding areal differentiation in geography, where we study how phenomena vary across different spaces Fundamentals of Physical Geography, Geography Class XI, p.11.

In a three-dimensional (3D) system, any point P is represented as (x, y, z). These three values represent the signed distances of the point from the origin along each respective axis. For instance, the 'x' coordinate tells you how far to move forward or backward along the X-axis, 'y' tells you how far to move left or right, and 'z' tells you the height or depth. This precise measurement is the same logic used when we draw perpendiculars to surfaces to calculate the angle of incidence or refraction in optics, as we do when studying how light moves through a triangular prism Science, Class X, p.165.

A crucial feature of this system is that the axes are orthogonal, meaning they meet at 90° angles. This allows us to use the Pythagorean theorem to find the direct distance from the origin to any point. If a point has coordinates (x, y, z), its distance d from the origin is calculated as d = √(x² + y² + z²). By breaking down a position into these three components, we can analyze complex motions or physical structures by looking at what happens along each axis independently.

Sources: Fundamentals of Physical Geography, Geography Class XI, Geography as a Discipline, p.11; Science, Class X, The Human Eye and the Colourful World, p.165

2. Principles of Vector Resolution and Projections (basic)

At its heart, Vector Resolution is the process of breaking a single vector into two or more components that act in specific directions (usually along the x, y, and z axes). Think of it like describing a treasure map: instead of saying "walk 100 meters northeast," you might say "walk 70 meters East and 70 meters North." Both get you to the same spot, but the latter is much easier to calculate with. This principle is fundamental in physics, whether we are analyzing the path of light rays Science-Class VII . NCERT(Revised ed 2025), Light: Shadows and Reflections, p.168 or the forces acting on a principal axis Science , class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.136.

When we move into three-dimensional space, we often use angles to define a point's position. Imagine a vector of length l starting at the origin. If we know the polar angle (q)—the angle it makes with the vertical z-axis—we can find its "shadow" or projection on the flat ground (the xy-plane). Using basic trigonometry, this projection length is l sin(q). Once we have this projection on the ground, we can resolve it further into x and y components using a second angle, the azimuthal angle (a), measured from the x-axis.

To find the y-component, we take that horizontal projection (l sin(q)) and multiply it by the sine of the azimuthal angle (a). This gives us the final formula: y = l sin(q) sin(a). This multi-step resolution allows us to translate complex 3D directions into simple Cartesian coordinates (x, y, z) that are easier to use in mechanical equations.

Feature	2D Resolution	3D Resolution (Spherical)
Components	Horizontal (x) and Vertical (y)	x, y, and z components
Angles Used	One angle (θ)	Polar angle (q) and Azimuthal angle (a)
Y-Component	y = R sin(θ)	y = l sin(q) sin(a)

Sources: Science-Class VII . NCERT(Revised ed 2025), Light: Shadows and Reflections, p.168; Science , class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.136

3. Geographical Mapping: Latitude and Longitude (intermediate)

To navigate or locate any point on our spherical Earth, we use a sophisticated grid system known as Geographical Mapping. Imagine the Earth not as a flat map, but as a three-dimensional sphere. To pinpoint a location (Point P), we rely on two primary angular measurements: Latitude and Longitude. These are essentially imaginary lines that allow us to define any position with precision, such as New Delhi being located at approximately 28° N and 77° E PMF IAS, Physical Geography, p.240.

Latitude is defined as the angular distance of a point measured north or south of the Equator (0°). Because these lines run parallel to the Equator and never meet, they are often called Parallels PMF IAS, Physical Geography, p.250. While the Equator is the longest parallel, the circles become progressively smaller as they approach the North Pole (90° N) and South Pole (90° S). Significant latitudes include the Tropic of Cancer (23.5° N), Tropic of Capricorn (23.5° S), and the Arctic/Antarctic Circles (66.5° N/S) PMF IAS, Physical Geography, p.250. These lines are critical for understanding climate zones and the angle of the sun's rays.

Longitude, on the other hand, represents the angular distance measured along the Equator, east or west of the Prime Meridian (0°). These lines are called Meridians. Unlike latitudes, all meridians are equal in length and converge at the poles PMF IAS, Physical Geography, p.250. The system is split into 180° East and 180° West; interestingly, the 180°W and 180°E lines are actually the same longitude, known as the International Date Line NCERT Class VI, Exploring Society, p.16.

Feature	Latitudes (Parallels)	Longitudes (Meridians)
Reference Line	Equator (0°)	Prime Meridian (0°)
Direction	North or South	East or West
Line Length	Decreases toward poles	All lines are equal in length
Maximum Value	90° (at the Poles)	180° (at the Date Line)

In advanced mechanics and navigation, these coordinates act as spherical polar coordinates. By knowing the radius of the Earth (the distance from the center) and the two angles (latitude and longitude), we can mathematically project a point on the surface into a three-dimensional Cartesian space (x, y, z). This conversion is vital for satellite technology, aviation, and even understanding Earth's magnetic field, where the magnetic dip varies from 0° at the magnetic equator to 90° at the magnetic poles PMF IAS, Physical Geography, p.77.

Sources: Physical Geography by PMF IAS, Latitudes and Longitudes, p.240; Physical Geography by PMF IAS, Latitudes and Longitudes, p.250; Exploring Society: India and Beyond. Social Science-Class VI. NCERT, Locating Places on the Earth, p.16; Physical Geography by PMF IAS, Earth's Magnetic Field, p.77

4. Science & Tech: Satellite Orbits and Positioning (intermediate)

To understand how we track a satellite in space, we first need to define its orbit. An orbit is the specific path an object takes while revolving around another body, such as a satellite around the Earth Science-Class VII . NCERT(Revised ed 2025), Earth, Moon, and the Sun, p.176. While these paths are often simplified as circles, they are governed by Kepler’s Laws, which state that orbits are actually ellipses with the central body (like the Sun or Earth) at one focus Physical Geography by PMF IAS, The Solar System, p.21. For a satellite to maintain its position without falling or drifting, it must reside in regions like the exosphere, where the air is thin enough to minimize atmospheric drag Physical Geography by PMF IAS, Earths Atmosphere, p.280.

Once a satellite is in orbit, we pinpoint its exact location using a Spherical Polar Coordinate System. Imagine a satellite at point P. Instead of just using flat grid lines, we use three parameters: the radial distance (l) from the Earth's center, a polar angle (q) measured down from the vertical Z-axis (usually the North Pole), and an azimuthal angle (a) measured horizontally from the X-axis (often the Prime Meridian). To find the satellite's position in a standard 3D grid (Cartesian coordinates), we must "resolve" these angles using trigonometry.

The process of finding the y-component (its position along the Y-axis) happens in two steps:

• Step 1: Projection onto the horizontal plane. We first find the "shadow" of the radial distance l on the flat XY-plane. This is calculated as l sin(q).
• Step 2: Resolution along the Y-axis. We then take this horizontal projection and find its component along the Y-axis by multiplying it by the sine of the horizontal angle a.

Therefore, the final formula for the y-position is y = l sin(q) sin(a). This mathematical precision is what allows ISRO to successfully place and track satellites like the INSAT or IRS series in their designated slots in space Geography of India, Majid Husain, Transport, Communications and Trade, p.56.

Sources: Science-Class VII . NCERT(Revised ed 2025), Earth, Moon, and the Sun, p.176; Physical Geography by PMF IAS, The Solar System, p.21; Physical Geography by PMF IAS, Earths Atmosphere, p.280; Geography of India ,Majid Husain, Transport, Communications and Trade, p.56

5. Defining Spherical Polar Geometry (l, q, a) (intermediate)

In mechanics and geography, we often need to describe a point's position in 3D space. While the standard Cartesian system (x, y, z) uses distances along straight axes, the Spherical Polar Coordinate system uses a distance and two angles. This is much like how we locate a city on Earth using its distance from the center and its angular coordinates—latitude and longitude Physical Geography by PMF IAS, Latitudes and Longitudes, p.250.

In this specific system, we define the position of a point P using three variables: l (the radial distance from the origin), q (the polar angle), and a (the azimuthal angle). Understanding how these translate to Cartesian coordinates is a fundamental skill in vector mechanics:

• Radial Distance (l): This is the straight-line distance from the origin to point P.
• Polar Angle (q): This is the angle measured downward from the positive z-axis (the "North Pole"). It is conceptually similar to magnetic inclination, which measures the angle a needle makes with the horizontal plane Physical Geography by PMF IAS, Earths Magnetic Field, p.77.
• Azimuthal Angle (a): This is the angle measured in the horizontal xy-plane, starting from the positive x-axis.

To find the y-component of point P, we use a two-step geometric resolution. First, we project the radial distance l onto the horizontal xy-plane; this "shadow" has a length of l sin(q). Next, we resolve this shadow along the y-axis. Since the angle a is measured from the x-axis, the y-component (the opposite side of the triangle in the xy-plane) is found using the sine of a. Therefore, the coordinate is expressed as y = l sin(q) sin(a).

Cartesian Component	Spherical Formula	Geometric Logic
x	l sin(q) cos(a)	Projection on xy-plane resolved along the x-axis.
y	l sin(q) sin(a)	Projection on xy-plane resolved along the y-axis.
z	l cos(q)	Direct vertical resolution along the z-axis.

Sources: Physical Geography by PMF IAS, Latitudes and Longitudes, p.250; Physical Geography by PMF IAS, Earths Magnetic Field, p.77

6. Mathematical Transformations: Spherical to Cartesian (exam-level)

In mechanics and advanced physics, we often find that the Cartesian coordinate system (x, y, z) isn't the most efficient way to describe motion, especially when dealing with objects moving around a central point. This is where the Spherical Polar Coordinate System becomes invaluable. Instead of thinking in terms of 'left-right' or 'up-down' grids, we define a point P using a radial distance l from the origin, a polar angle q (measured from the vertical z-axis), and an azimuthal angle a (measured from the x-axis in the horizontal plane). This logic of establishing an origin and measuring distances and angles is a sophisticated extension of the New Cartesian Sign Convention used to locate images in optics Science, Light – Reflection and Refraction, p.142.

To transform these spherical coordinates back into Cartesian values, we use a two-step geometric projection. Imagine a line of length l tilting away from the vertical z-axis by an angle q. To find how much of this length 'lives' in the horizontal xy-plane, we calculate its projection. Using basic trigonometry, the 'shadow' of this line on the ground (the xy-plane) has a length of l sin(q). This is the crucial first step: resolving the 3D distance into a 2D plane.

Once we have this horizontal projection, l sin(q), we treat it like a 2D vector on the floor. Since the azimuthal angle a tells us how far this projection has rotated away from the x-axis toward the y-axis, we resolve it one more time. The component along the y-axis is found by multiplying this projection by the sine of the azimuthal angle. Therefore, the transformation formula for the y-component is y = l sin(q) sin(a). This systematic approach ensures that whether we are calculating the position of a planet or the path of a light ray through a lens, our spatial coordinates remain mathematically consistent Science, Light – Reflection and Refraction, p.155.

Sources: Science, Light – Reflection and Refraction, p.142; Science, Light – Reflection and Refraction, p.155; Certificate Physical and Human Geography, The Earth's Crust, p.9

7. Solving the Original PYQ (exam-level)

Now that you have mastered the fundamental transformations between coordinate systems, this question tests your ability to apply those geometric projections in a 3D space. The "building blocks" you recently studied—specifically the resolution of a vector into its components—are directly applicable here. Although the question uses the notation (l, q, a) instead of the standard (r, θ, φ), the logic remains the same: you must first project the radial distance P onto the xy-plane and then further resolve that projection onto the y-axis. As per the principles found in Mathematical Methods for Physicists by Arfken, the projection on the plane is P sin(q), and since the azimuthal angle a starts from the x-axis, the component perpendicular to it (the y-axis) must use the sine function.

To arrive at the correct answer, follow this mental walkthrough: visualize the angle q pulling the vector away from the z-axis toward the floor (xy-plane); this gives you the "shadow" length P sin(q). Next, visualize that shadow on the floor rotating by angle a away from the x-axis toward the y-axis. Since the y-axis is the "opposite" side relative to the x-axis starting point, we multiply the shadow length by sin(a). This leads us directly to (A) Psinq sina. This systematic approach ensures you aren't just memorizing formulas but are mapping the spatial orientation of the point accurately.

UPSC frequently uses distractor options to catch students who mix up their trigonometric assignments. Option (B) is a classic trap; it represents the x-component because it uses cosa. Options (C) and (D) are even more dangerous as they lead with cosq, which would actually define the z-component (projection onto the vertical axis). Remember, in these problems, the polar angle q determines the height, while the azimuthal angle a determines the orientation on the ground. Confusing these is the most common path to an incorrect choice.