Introduction to 3D Geometry Class 11 Notes: Formulas, Concepts & Examples

Key Terms in 3D Geometry

Coordinate Axes

Three mutually perpendicular lines (X'OX, Y'OY, Z'OZ) that intersect at the origin O.

Coordinate Planes

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

Coordinates of a Point

An ordered triplet (x, y, z) that specifies the position of a point in 3D space, representing its perpendicular distances from the YZ, XZ, and XY planes, respectively.

Octants

The eight regions into which the three coordinate planes divide the 3D space. The sign of the coordinates (x, y, z) determines the octant.

Origin

The point of intersection of the three coordinate axes, with coordinates (0, 0, 0).

Collinear Points

Three or more points that lie on the same straight line.

Centroid of a Triangle

The point of concurrency of the medians of a triangle. Its coordinates are the average of the coordinates of the vertices.

Understanding Coordinate System in 3D Space

In two-dimensional geometry, we use a pair of coordinates (x, y) to locate a point on a plane. To locate a point in three-dimensional space, we need an additional reference. We use three mutually perpendicular lines, known as the X-axis, Y-axis, and Z-axis. These axes intersect at a single point called the Origin (O). The coordinates of the origin are (0, 0, 0). Together, these axes form the rectangular coordinate system. The planes formed by taking the axes in pairs are called coordinate planes.

• The XY-plane is the plane containing the X and Y axes. For any point on this plane, its z-coordinate is zero. The equation of the XY-plane is z = 0.
• The YZ-plane contains the Y and Z axes. For any point on this plane, its x-coordinate is zero. The equation is x = 0.
• The XZ-plane contains the X and Z axes. For any point on this plane, its y-coordinate is zero. The equation is y = 0.

These three planes divide the space into eight parts called octants. The coordinates of a point P(x, y, z) represent its signed perpendicular distances from the three coordinate planes. For instance, 'x' is the distance from the YZ-plane, 'y' is the distance from the XZ-plane, and 'z' is the distance from the XY-plane.

Key Formulas & Concepts

• Distance Formula: The distance between two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) is given by PQ = √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²].
• Distance from Origin: The distance of a point P(x, y, z) from the origin O(0,0,0) is OP = √(x² + y² + z²).
• Section Formula (Internal Division): If R divides PQ in the ratio m:n internally, then R = ( (mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n), (mz₂+nz₁)/(m+n) ).
• Section Formula (External Division): If R divides PQ in the ratio m:n externally, then R = ( (mx₂-nx₁)/(m-n), (my₂-ny₁)/(m-n), (mz₂-nz₁)/(m-n) ).
• Mid-Point Formula: The midpoint of the line segment joining P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) is ( (x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2 ).
• Centroid of a Triangle: The centroid of a triangle with vertices (x₁, y₁, z₁), (x₂, y₂, z₂), and (x₃, y₃, z₃) is ( (x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3, (z₁+z₂+z₃)/3 ).
• Coordinates on Axes: A point on the X-axis is (x, 0, 0), on the Y-axis is (0, y, 0), and on the Z-axis is (0, 0, z).
• Coordinates on Planes: A point on the XY-plane is (x, y, 0), on the YZ-plane is (0, y, z), and on the XZ-plane is (x, 0, z).
• Test for Collinearity: Three points A, B, and C are collinear if the sum of the lengths of any two line segments is equal to the length of the third (e.g., AB + BC = AC).

2D Geometry vs. 3D Geometry

Worked Examples

• {"example":"Find the distance between the points P(1, -3, 4) and Q(-4, 1, 2).","solution":"Using the distance formula, PQ = √[(-4-1)² + (1-(-3))² + (2-4)²] = √[(-5)² + (4)² + (-2)²] = √[25 + 16 + 4] = √45 = 3√5 units."}
• {"example":"Find the coordinates of the point which divides the line segment joining points A(2, -4, 3) and B(-4, 5, -6) in the ratio 2:1 internally.","solution":"Using the internal section formula with m=2, n=1: x = (2(-4) + 12)/(2+1) = -6/3 = -2. y = (25 + 1(-4))/(2+1) = 6/3 = 2. z = (2(-6) + 13)/(2+1) = -9/3 = -3. The coordinates are (-2, 2, -3)."}
• {"example":"Find the centroid of a triangle whose vertices are A(1, 2, 3), B(2, 3, 4), and C(3, 4, 5).","solution":"Using the centroid formula: G = ( (1+2+3)/3, (2+3+4)/3, (3+4+5)/3 ) = (6/3, 9/3, 12/3) = (2, 3, 4)."}

Exam Traps & Tips

A common trap is mixing up the section formulas. For internal division, the ratio m:n leads to m+n in the denominator. For external division, it's m-n. Always double-check which one the question asks for. When asked to find the ratio in which a plane divides a line segment, assume the ratio is k:1. Solve for 'k'. If k is positive, the division is internal. If k is negative, the division is external. Another frequent error is sign mismanagement in the distance formula. Remember that (x₂ - x₁)² is always non-negative, even if x₂ - x₁ is negative.

Practice Questions with Solutions

• In which octant does the point (2, -4, -7) lie? Eighth octant (x: +, y: -, z: -).
• What are the coordinates of a point on the Z-axis at a distance of 5 units from the origin in the negative direction? (0, 0, -5).
• State the formula for the midpoint of a line segment joining P(x₁, y₁, z₁) and Q(x₂, y₂, z₂). M = ( (x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2 ).
• Find the ratio in which the YZ-plane divides the line segment joining the points (-2, 4, 7) and (3, -5, 8). Let the ratio be k:1. The x-coordinate of the dividing point is (3k - 2)/(k+1). Since the point lies on the YZ-plane, its x-coordinate is 0. So, 3k - 2 = 0, which gives k = 2/3. The ratio is 2:3 internally.

Frequently Asked Questions

Frequently Asked Questions