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Introduction to Three-Dimensional Geometry - Coordinate Axes and Coordinate Planes in 3D

Grade 11ICSE

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

🔑Concepts

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The Coordinate Axes consist of three mutually perpendicular lines, denoted as the XX, YY, and ZZ axes, which intersect at a single point called the Origin O(0,0,0)O(0, 0, 0). Visually, these axes represent the three dimensions of space: length, width, and height, similar to the three edges meeting at the bottom corner of a room.

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The three Coordinate Planes are formed by the axes taken in pairs. The XYXY-plane contains the XX and YY axes, the YZYZ-plane contains the YY and ZZ axes, and the ZXZX-plane contains the ZZ and XX axes. Visually, if you imagine a room, the floor represents the XYXY-plane, while two adjacent walls represent the YZYZ and ZXZX planes.

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Space is divided into eight regions by these three coordinate planes, known as Octants. This is an extension of the four quadrants in 2D geometry. The octant in which a point lies is determined by the positive or negative signs of its (x,y,z)(x, y, z) coordinates.

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The coordinates of a point P(x,y,z)P(x, y, z) represent its signed perpendicular distances from the coordinate planes. Specifically, xx is the distance from the YZYZ-plane, yy is the distance from the ZXZX-plane, and zz is the distance from the XYXY-plane. Visually, to reach point PP, you walk xx units along the XX-axis, turn and walk yy units parallel to the YY-axis, and then move zz units up or down.

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Points residing on the axes have a specific coordinate structure where two coordinates are always zero. A point on the XX-axis is (x,0,0)(x, 0, 0), a point on the YY-axis is (0,y,0)(0, y, 0), and a point on the ZZ-axis is (0,0,z)(0, 0, z). Similarly, points on a coordinate plane have at least one zero coordinate; for example, any point on the XYXY-plane is (x,y,0)(x, y, 0).

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The Sign Convention for Octants follows a specific pattern. Octants I, II, III, and IV all have a positive zz-coordinate (z>0z > 0) and follow the signs of the 2D quadrants for xx and yy. Octants V, VI, VII, and VIII have a negative zz-coordinate (z<0z < 0) and repeat the xx and yy sign patterns of the first four octants.

📐Formulae

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

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

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

Coordinates of a point on the XX-axis: (x,0,0)(x, 0, 0)

Coordinates of a point on the YY-axis: (0,y,0)(0, y, 0)

Coordinates of a point on the ZZ-axis: (0,0,z)(0, 0, z)

Distance of point P(x,y,z)P(x, y, z) from the origin: d=sqrtx2+y2+z2d = \\sqrt{x^2 + y^2 + z^2}

Distance between two points P1(x1,y1,z1)P_1(x_1, y_1, z_1) and P2(x2,y2,z2)P_2(x_2, y_2, z_2): d=sqrt(x2−x1)2+(y2−y1)2+(z2−z1)2d = \\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}

💡Examples

Problem 1:

Determine the octants in which the following points lie: A(2,−3,5)A(2, -3, 5) and B(−1,−4,−2)B(-1, -4, -2).

Solution:

  1. For point A(2,−3,5)A(2, -3, 5): Here xx is positive (++), yy is negative (−-), and zz is positive (++). Looking at the sign convention, the (+,−,+)(+, -, +) pattern corresponds to Octant IV.
  2. For point B(−1,−4,−2)B(-1, -4, -2): Here xx is negative (−-), yy is negative (−-), and zz is negative (−-). The (−,−,−)(- , - , -) pattern corresponds to Octant VII.

Explanation:

To identify the octant, we look at the signs of the x,y,zx, y, z coordinates. Octants I-IV have z>0z > 0 and Octants V-VIII have z<0z < 0.

Problem 2:

Find the distance of the point P(3,−4,12)P(3, -4, 12) from (i) the XYXY-plane and (ii) the Origin.

Solution:

  1. Distance from the XYXY-plane: The perpendicular distance of any point (x,y,z)(x, y, z) from the XYXY-plane is given by ∣z∣|z|. Here, z=12z = 12, so the distance is ∣12∣=12|12| = 12 units.
  2. Distance from the Origin O(0,0,0)O(0, 0, 0): Using the distance formula d=sqrtx2+y2+z2d = \\sqrt{x^2 + y^2 + z^2}, we get: d=sqrt32+(−4)2+122d = \\sqrt{3^2 + (-4)^2 + 12^2} d=sqrt9+16+144d = \\sqrt{9 + 16 + 144} d=sqrt169=13d = \\sqrt{169} = 13 units.

Explanation:

The distance from a coordinate plane is the absolute value of the 'missing' coordinate. The distance from the origin uses the 3D version of the Pythagorean theorem.