Introduction To Three Dimensional Geometry Class 11 Chapter Notes

Key Definitions

Coordinate Axes

Three mutually perpendicular lines (X-axis, Y-axis, Z-axis) that intersect at a common point called the origin, forming a 3D Cartesian coordinate system.

Origin

The point of intersection of the three coordinate axes in 3D space, denoted as O(0, 0, 0).

Coordinate Planes

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

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 in Space

An ordered triplet (x, y, z) that uniquely represents the position of a point in three-dimensional space, where x, y, and z are the perpendicular distances from the YZ, ZX, and XY planes, respectively.

Distance Formula (3D)

A formula used to calculate the distance between two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) in 3D space: d = √((x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²) .

Section Formula (3D)

A formula used to find the coordinates of a point that divides a line segment joining two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) in a given ratio m:n, either internally or externally.

The 3D Coordinate System and Octants

Just as a point in a 2D plane is defined by two coordinates (x, y), a point in three-dimensional space is defined by three coordinates (x, y, z). This system is built upon three mutually perpendicular lines called the coordinate axes: the X-axis, Y-axis, and Z-axis. These axes intersect at a single point known as the origin, denoted as O(0, 0, 0).

These three axes define three mutually perpendicular coordinate planes:

1. The XY-plane (where z=0)
2. The YZ-plane (where x=0)
3. The ZX-plane (where y=0)

These three planes collectively divide the entire space into eight regions, each called an octant. The sign of the coordinates (x, y, z) determines which octant a point lies in. For example, the first octant (usually denoted I) is where x, y, and z are all positive (+, +, +). The other octants follow specific sign conventions, which are important for visualizing point locations:

• I Octant: (+, +, +)
• II Octant: (-, +, +)
• III Octant: (-, -, +)
• IV Octant: (+, -, +)
• V Octant: (+, +, -)
• VI Octant: (-, +, -)
• VII Octant: (-, -, -)
• VIII Octant: (+, -, -)

Understanding these octants helps in accurately plotting and interpreting points in 3D space, which is fundamental to solving problems in three-dimensional geometry.

Key Formulas and Must Remember Facts

• Distance between two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂): PQ = √((x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²)
• Distance of a point P(x, y, z) from the origin O(0, 0, 0): OP = √(x² + y² + z²)
• Distance of a point P(x, y, z) from the X-axis: √(y² + z²)
• Distance of a point P(x, y, z) from the Y-axis: √(x² + z²)
• Distance of a point P(x, y, z) from the Z-axis: √(x² + y²)
• Distance of a point P(x, y, z) from the XY-plane: |z|
• Distance of a point P(x, y, z) from the YZ-plane: |x|
• Distance of a point P(x, y, z) from the ZX-plane: |y|
• Section Formula (Internal Division): The coordinates of a point R that divides the line segment joining P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) internally in the ratio m:n are: R = ((nx₁ + mx₂)/(m+n), (ny₁ + my₂)/(m+n), (nz₁ + mz₂)/(m+n))
• Section Formula (External Division): The coordinates of a point R that divides the line segment joining P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) externally in the ratio m:n are: R = ((nx₁ - mx₂)/(n-m), (ny₁ - my₂)/(n-m), (nz₁ - mz₂)/(n-m))
• Mid-point Formula: A special case of the internal section formula where m=n=1. The midpoint M of P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) is: M = ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2)
• Centroid of a Triangle: For a triangle with vertices (x₁,y₁,z₁), (x₂,y₂,z₂), (x₃,y₃,z₃), the centroid is: ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3, (z₁+z₂+z₃)/3) .

Worked Examples

• Example 1: Distance between two points Find the distance between the points A(2, 3, 5) and B(4, 3, 1). Solution: Using the distance formula d = √((x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²) d = √((4-2)² + (3-3)² + (1-5)²) d = √(2² + 0² + (-4)²) d = √(4 + 0 + 16) d = √20 = 2√5 units.
• Example 2: Section Formula (Internal Division) Find the coordinates of the point that divides the line segment joining P(–2, 3, 5) and Q(1, –4, 6) internally in the ratio 2:3. Solution: Here, (x₁, y₁, z₁) = (–2, 3, 5), (x₂, y₂, z₂) = (1, –4, 6), and m:n = 2:3. Using the internal section formula: x = (nx₁ + mx₂)/(m+n) = (3(-2) + 2(1))/(2+3) = (-6+2)/5 = -4/5 y = (ny₁ + my₂)/(m+n) = (3(3) + 2(-4))/(2+3) = (9-8)/5 = 1/5 z = (nz₁ + mz₂)/(m+n) = (3(5) + 2(6))/(2+3) = (15+12)/5 = 27/5 Therefore, the required point is (-4/5, 1/5, 27/5) .

Exam Tip: Avoiding Common Traps

When solving problems in 3D geometry, pay close attention to the following:

• Sign Errors: A common mistake is mismanaging negative signs, especially in the distance and section formulas. Double-check your calculations, particularly with subtraction and squaring.
• Octant Identification: Carefully determine the signs of x, y, and z coordinates to correctly identify the octant. Remember the order (x, y, z) for sign conventions.
• Formula Recall: Ensure you remember the correct form of the distance formula and both internal/external section formulas. A misplaced sign or variable can lead to an incorrect answer.
• Understanding Plane vs. Axis Distances: Differentiate between the distance of a point from an axis and its distance from a coordinate plane. For instance, distance from XY-plane is |z|, but from X-axis is √(y² + z²) .

Practice Questions with Solutions

• Q: In which octant does the point P(4, -2, -3) lie? A: The coordinates are (+, -, -). This corresponds to the VIII octant.
• Q: What is the distance of the point (3, 4, 5) from the YZ-plane? A: The distance of a point (x, y, z) from the YZ-plane is |x|. So, the distance is |3| = 3 units.
• Q: Find the coordinates of the midpoint of the line segment joining A(1, -1, 3) and B(-3, 4, -5). A: Using the midpoint formula ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2) : ((1-3)/2, (-1+4)/2, (3-5)/2) = (-2/2, 3/2, -2/2) = (-1, 3/2, -1) .
• Q: If a point lies on the Z-axis, what are its x and y coordinates? A: If a point lies on the Z-axis, its x and y coordinates are both zero. The point will be of the form (0, 0, z).

Frequently Asked Questions