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Introduction to Three Dimensional Geometry - Coordinate Axes and Coordinate Planes in Three Dimensional Space

Grade 11CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Three Mutually Perpendicular Axes: In three-dimensional geometry, we use three mutually perpendicular lines passing through a common point called the origin O(0,0,0)O(0,0,0). These lines are the XX-axis, YY-axis, and ZZ-axis. Visually, you can imagine the corner of a room where the floor meets two walls; the three edges meeting at that corner represent these three axes.

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Coordinate Planes: The three axes taken in pairs determine three coordinate planes: the XYXY-plane (containing the XX and YY axes), the YZYZ-plane (containing the YY and ZZ axes), and the ZXZX-plane (containing the ZZ and XX axes). Visually, if the XYXY-plane is the floor, then the YZYZ and ZXZX planes are the two adjacent walls.

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Octants: The three coordinate planes divide the entire space into eight distinct regions called octants. These are named as XOYZX O Y Z, X′OYZX' O Y Z, X′OY′ZX' O Y' Z, XOY′ZX O Y' Z, XOYZ′X O Y Z', X′OYZ′X' O Y Z', X′OY′Z′X' O Y' Z', and XOY′Z′X O Y' Z'. Visually, this is similar to how a horizontal and vertical cut divides a cake into 4 quadrants, but adding a third horizontal slice through the middle creates 8 pieces.

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Coordinates of a Point: Every point PP in space is associated with an ordered triplet (x,y,z)(x, y, z). The xx-coordinate is the perpendicular distance from the YZYZ-plane, the yy-coordinate is the perpendicular distance from the ZXZX-plane, and the zz-coordinate is the perpendicular distance from the XYXY-plane.

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Points on Axes and Planes: If a point lies on the XX-axis, its coordinates are of the form (x,0,0)(x, 0, 0). If it lies in the XYXY-plane, its coordinates are (x,y,0)(x, y, 0). Visually, a point on an axis 'sticks' to that single line, while a point in a plane stays flat against that specific surface without moving 'out' into the third dimension.

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Sign Convention in Octants: The sign of the coordinates (x,y,z)(x, y, z) determines which octant a point belongs to. For example, in Octant I, all coordinates are positive (+,+,+)(+,+,+). In Octant VII, all coordinates are negative (−,−,−)(-,-,-). The signs follow a specific pattern based on the orientation relative to the origin.

📐Formulae

Coordinates of the origin: (0,0,0)(0, 0, 0) contrast to 2D (0,0)(0, 0)

Equation of the XYXY-plane: z=0z = 0

Equation of the YZYZ-plane: x=0x = 0

Equation of the ZXZX-plane: y=0y = 0

Equation of the XX-axis: y=0,z=0y = 0, z = 0

Equation of the YY-axis: x=0,z=0x = 0, z = 0

Equation of the ZZ-axis: x=0,y=0x = 0, y = 0

Distance of a point P(x,y,z)P(x, y, z) from the origin O(0,0,0)O(0, 0, 0): OP=x2+y2+z2OP = \sqrt{x^2 + y^2 + z^2}

Distance between two points P(x1,y1,z1)P(x_1, y_1, z_1) and Q(x2,y2,z2)Q(x_2, y_2, z_2): PQ=(x2−x1)2+(y2−y1)2+(z2−z1)2PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}

💡Examples

Problem 1:

Determine the octant in which the following points lie: (i) (4,−2,3)(4, -2, 3) and (ii) (−4,2,−5)(-4, 2, -5).

Solution:

For point (i) (4,−2,3)(4, -2, 3), we have x>0x > 0 (positive), y<0y < 0 (negative), and z>0z > 0 (positive). This corresponds to Octant IV (XOY′ZXOY'Z). For point (ii) (−4,2,−5)(-4, 2, -5), we have x<0x < 0 (negative), y>0y > 0 (positive), and z<0z < 0 (negative). This corresponds to Octant VI (X′OYZ′X'OYZ').

Explanation:

To identify the octant, check the signs of the x,y,x, y, and zz coordinates. There are 8 combinations: (+,+,+) is I, (-,+,+) is II, (-,-,+) is III, (+,-,+) is IV. Adding a negative zz maps to V, VI, VII, VIII respectively.

Problem 2:

A point PP is at a distance of 55 units from the XX-axis and lies on the YY-axis. What are its coordinates?

Solution:

Since the point lies on the YY-axis, its xx and zz coordinates must be zero. Thus, the point is of the form (0,y,0)(0, y, 0). The distance from the XX-axis to a point (x,y,z)(x, y, z) is given by y2+z2\sqrt{y^2 + z^2}. Here, y2+02=5\sqrt{y^2 + 0^2} = 5, which means ∣y∣=5|y| = 5. Therefore, the coordinates are (0,5,0)(0, 5, 0) or (0,−5,0)(0, -5, 0).

Explanation:

We use the property that points on the YY-axis have x=0x=0 and z=0z=0, and then apply the geometric definition of distance from an axis.