CBSE Class 11 Maths: Introduction to 3D Geometry - Exercise 12.1

Understanding the 3D Coordinate System

In two-dimensional geometry, we use two perpendicular lines, the X-axis and the Y-axis, intersecting at the origin (0,0) to define the position of a point (x,y) on a plane. Three-dimensional geometry extends this idea by introducing a third axis, the Z-axis, which is perpendicular to both the X-axis and the Y-axis at their point of intersection, the origin (0,0,0). Imagine the corner of a room: the floor defines the XY-plane, and the two walls intersecting at that corner represent the XZ-plane and YZ-plane, respectively. The three axes (X, Y, Z) are mutually perpendicular to each other, forming a right-handed coordinate system. A point in 3D space is uniquely represented by an ordered triplet (x, y, z), where 'x' is its distance from the YZ-plane, 'y' is its distance from the XZ-plane, and 'z' is its distance from the XY-plane. The sign of each coordinate indicates its direction along the respective axis from the origin. For instance, a positive 'x' means it's on the positive X-axis side, and a negative 'z' means it's below the XY-plane.

Key Definitions: Coordinate Planes and Octants

3D Coordinate System

A system that uses three mutually perpendicular axes (X, Y, and Z) intersecting at a common origin to define the position of any point in three-dimensional space by an ordered triplet (x, y, z).

Coordinate Planes

The three planes formed by the intersection of the coordinate axes: the XY-plane (z=0), the YZ-plane (x=0), and the ZX-plane (y=0). These planes divide the space into eight regions.

Origin (O)

The point where the three coordinate axes intersect. Its coordinates are (0, 0, 0).

Octants

The eight regions into which the three coordinate planes divide the entire 3D space. Each octant is defined by the signs of the x, y, and z coordinates.

Coordinates of a Point (x, y, z)

The directed perpendicular distances of a point from the YZ-plane, XZ-plane, and XY-plane, respectively. The signs indicate the octant or direction.

How to Identify Octants

1. Understand the Axes and Planes — Visualize the X, Y, and Z axes. The positive directions are typically shown pointing outwards/upwards. Remember that the XY-plane is where z=0, YZ-plane is where x=0, and XZ-plane is where y=0.
2. Note the Signs of Coordinates — For any given point (x, y, z), observe the sign of each coordinate (positive or negative). These signs directly determine which octant the point lies in.
3. Relate Signs to Octant Number — There are eight octants, numbered I to VIII, based on the combination of signs: Octant I: (+, +, +) Octant II: (–, +, +) Octant III: (–, –, +) Octant IV: (+, –, +) Octant V: (+, +, –) Octant VI: (–, +, –) Octant VII: (–, –, –) Octant VIII: (+, –, –) It's helpful to remember that octants I-IV are above the XY-plane (z>0) and octants V-VIII are below it (z<0).
4. Apply for Points on Axes or Planes — If a point lies on an axis, two of its coordinates will be zero (e.g., (x, 0, 0) for X-axis). If it lies on a coordinate plane, one coordinate will be zero (e.g., (x, y, 0) for XY-plane). Points on axes or planes do not lie in any specific octant; they lie on the boundary between them.

Worked Examples from Ex 12.1 Concepts

• Example 1: Identifying Octants Identify the octant in which the following points lie: (i) (2, 3, 1) (ii) (-4, 2, 5) (iii) (1, -6, 3) (iv) (-2, -3, -4) Solution: (i) For point (2, 3, 1): x > 0, y > 0, z > 0. This combination corresponds to Octant I. (ii) For point (-4, 2, 5): x < 0, y > 0, z > 0. This combination corresponds to Octant II. (iii) For point (1, -6, 3): x > 0, y < 0, z > 0. This combination corresponds to Octant IV. (iv) For point (-2, -3, -4): x < 0, y < 0, z < 0. This combination corresponds to Octant VII. Example 2: Points on Coordinate Planes/Axes Which coordinate plane(s) do the following points lie on? (i) (5, 0, 3) (ii) (0, -2, 0) (iii) (-1, 7, 0) Solution: (i) For point (5, 0, 3): The y-coordinate is 0. Points with y=0 lie on the XZ-plane. (ii) For point (0, -2, 0): The x-coordinate and z-coordinate are 0. Points of the form (0, y, 0) lie on the Y-axis. (iii) For point (-1, 7, 0): The z-coordinate is 0. Points with z=0 lie on the XY-plane.
• Example 3: Determining Coordinates from Distances A point is 5 units away from the YZ-plane, 2 units away from the XZ-plane, and 4 units away from the XY-plane. If it lies in Octant VIII, find its coordinates. Solution: Step 1: Understand the meaning of distances from planes. Distance from YZ-plane = |x-coordinate| Distance from XZ-plane = |y-coordinate| * Distance from XY-plane = |z-coordinate| So, |x|=5, |y|=2, |z|=4. Step 2: Use the octant information to determine signs. Octant VIII has signs (+, -, -). This means x > 0, y < 0, z < 0. Step 3: Combine distances and signs to get coordinates. x = +5 (since x>0 and |x|=5) y = -2 (since y<0 and |y|=2) z = -4 (since z<0 and |z|=4) Final Answer: The coordinates of the point are (5, -2, -4).

Exam Tip: Avoiding Common Mistakes in 3D Geometry

Many students get confused with the sign conventions of octants. A handy trick is to first determine if the point is above or below the XY-plane (based on 'z'). If z > 0, it's one of Octants I-IV. If z < 0, it's one of Octants V-VIII. Then, apply your 2D quadrant knowledge to the x and y coordinates. For example, if z > 0 and (x, y) is in the first quadrant (+, +), it's Octant I. If (x, y) is in the fourth quadrant (+, -) with z > 0, it's Octant IV. Remember that points lying on any coordinate axis or coordinate plane are not considered to be in any specific octant; they lie on the boundaries between them. Always double-check the signs and the question carefully, especially when asked about points on a specific plane (e.g., on the XY-plane, the z-coordinate must be zero).

Practice Questions with Solutions

• Q: Name the octant in which the following points lie: (i) (1, 2, 3) (ii) (4, -2, 3) (iii) (-1, -2, 4) (iv) (4, -2, -5) (v) (-3, 1, -6) (vi) (2, -4, -7) A: Step 1: Analyze the signs of the x, y, and z coordinates for each point. (i) (1, 2, 3): x>0, y>0, z>0 (ii) (4, -2, 3): x>0, y<0, z>0 (iii) (-1, -2, 4): x<0, y<0, z>0 (iv) (4, -2, -5): x>0, y<0, z<0 (v) (-3, 1, -6): x<0, y>0, z<0 (vi) (2, -4, -7): x>0, y<0, z<0 Step 2: Match the sign combinations to the respective octants. (i) (+, +, +) -> Octant I (ii) (+, -, +) -> Octant IV (iii) (-, -, +) -> Octant III (iv) (+, -, -) -> Octant VIII (v) (-, +, -) -> Octant VI (vi) (+, -, -) -> Octant VIII Final answer: (i) Octant I, (ii) Octant IV, (iii) Octant III, (iv) Octant VIII, (v) Octant VI, (vi) Octant VIII.
• Q: Fill in the blanks: (i) The x-axis and y-axis taken together determine a plane known as the __________. (ii) The coordinates of points in the XY-plane are of the form __________. (iii) The coordinates of points on the Y-axis are of the form __________. A: Step 1: Recall the definitions of coordinate planes and axes. Step 2: For (i), the plane formed by the x and y axes is the XY-plane. Step 3: For (ii), any point on the XY-plane has its z-coordinate equal to zero. Step 4: For (iii), any point on the Y-axis has its x and z coordinates equal to zero. Final answer: (i) XY-plane, (ii) (x, y, 0), (iii) (0, y, 0).
• Q: For each of the following statements, determine whether it is true or false: (i) The YZ-plane is perpendicular to the X-axis. (ii) The coordinates of the origin are (0, 0, 0). (iii) A point (x, y, z) lies on the X-axis if x=0 and y=0. A: Step 1: Evaluate statement (i). The YZ-plane is defined by x=0. The X-axis is the line where y=0 and z=0. The X-axis passes through the origin and is perpendicular to the YZ-plane. Step 2: Evaluate statement (ii). The origin is the intersection of all three axes, so its coordinates are indeed (0, 0, 0). Step 3: Evaluate statement (iii). For a point to lie on the X-axis, its y and z coordinates must be zero, not x and y. If x=0 and y=0, the point lies on the Z-axis. Final answer: (i) True, (ii) True, (iii) False.
• Q: What are the coordinates of the foot of the perpendicular drawn from the point P(2, -3, 5) to the XY-plane? A: Step 1: Understand what it means to draw a perpendicular to the XY-plane. When a perpendicular is drawn from a point to the XY-plane, only the z-coordinate changes to 0, while the x and y coordinates remain the same. Step 2: Apply this concept to the given point P(2, -3, 5). The x-coordinate is 2, the y-coordinate is -3, and the z-coordinate becomes 0. Final answer: The coordinates of the foot of the perpendicular are (2, -3, 0).

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