This MCQ module is based on: Coordinate Axes and Coordinate Planes in 3D
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Coordinate Axes and Coordinate Planes in 3D
🎓 Class 11
Mathematics
CBSE
Theory
Ch 11 — Introduction to Three Dimensional Geometry
⏱ ~30 min
🌐 Language: [gtranslate]
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This mathematics assessment will be based on: Coordinate Axes and Coordinate Planes in 3D
Targeting Class 11 level in Coordinate Geometry, with Advanced difficulty.
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11.1 Introduction
You are already familiar with the two-dimensional Cartesian plane — a system that uses two perpendicular axes (the \(x\)-axis and \(y\)-axis) to locate any point in a plane using an ordered pair \((x,y)\). However, the world we inhabit is three-dimensional: the position of an aeroplane flying at some height above a runway, the location of a chandelier hanging in a hall, or the corner of a room cannot be described using just two numbers.
To locate such a point, we need three mutually perpendicular reference lines. This chapter extends the two-dimensional Cartesian system to three dimensions, introducing coordinate planes?, octants?, and the distance formula? in 3D space.
Real-World Motivation
Consider a room. The floor meeting two walls at a corner gives three mutually perpendicular edges. If a fly is sitting on the ceiling, its position can be specified by measuring three distances: how far along one wall, how far along the adjoining wall, and how high from the floor. These three numbers form the 3D coordinates of the fly.
Leonhard Euler
1707 – 1783
The Swiss mathematician Leonhard Euler made foundational contributions to analytic geometry in three dimensions. He systematised the treatment of surfaces and curves in space, introducing the modern notation \((x,y,z)\) for points in 3D and developing transformations between coordinate systems that are still taught today.
11.2 Coordinate Axes and Coordinate Planes in 3D
Imagine three mutually perpendicular lines \(X'OX\), \(Y'OY\), and \(Z'OZ\) intersecting at a common point \(O\). We call \(O\) the origin, and these three lines the coordinate axes:
- The line \(X'OX\) — the \(x\)-axis
- The line \(Y'OY\) — the \(y\)-axis
- The line \(Z'OZ\) — the \(z\)-axis
The positive direction on each axis is indicated by an arrow. The three axes, taken in pairs, determine three mutually perpendicular planes called the coordinate planes:
The Three Coordinate Planes
XY-plane: plane containing the \(x\)-axis and \(y\)-axis. Every point on it has \(z=0\).YZ-plane: plane containing the \(y\)-axis and \(z\)-axis. Every point on it has \(x=0\).
ZX-plane: plane containing the \(z\)-axis and \(x\)-axis. Every point on it has \(y=0\).
Fig 11.1: The three mutually perpendicular coordinate axes and the three coordinate planes dividing 3D space.
The Eight Octants
The three coordinate planes divide space into eight parts called octants?. Each octant is characterised by the signs of the coordinates of any point in it.
| Octant | I | II | III | IV | V | VI | VII | VIII |
|---|---|---|---|---|---|---|---|---|
| x | + | − | − | + | + | − | − | + |
| y | + | + | − | − | + | + | − | − |
| z | + | + | + | + | − | − | − | − |
For instance, the point \((3,4,5)\) lies in octant I (all positive); \((-2,3,1)\) lies in octant II; \((1,-2,-3)\) lies in octant VIII.
11.3 Coordinates of a Point in Space
Let \(P\) be any point in 3D space. Drop perpendiculars from \(P\) onto each of the three coordinate planes. The three signed distances so obtained are called the coordinates of \(P\), written as the ordered triple \((x,y,z)\).
Coordinates of a Point
If \(P(x,y,z)\) is a point in space, then:
- \(x\) = perpendicular distance of \(P\) from the YZ-plane (measured along \(x\)-axis)
- \(y\) = perpendicular distance of \(P\) from the ZX-plane (along \(y\)-axis)
- \(z\) = perpendicular distance of \(P\) from the XY-plane (along \(z\)-axis)
Fig 11.2: Point \(P(x,y,z)\) and its projections \(A\), \(B\), \(C\) on the axes and \(L\) on the XY-plane.
Coordinates of Special Points
- Origin \(O\): \((0,0,0)\)
- Any point on the \(x\)-axis has the form \((x,0,0)\)
- Any point on the \(y\)-axis has the form \((0,y,0)\)
- Any point on the \(z\)-axis has the form \((0,0,z)\)
- Any point on the XY-plane has \(z=0\), i.e., \((x,y,0)\)
- Any point on the YZ-plane has \(x=0\), i.e., \((0,y,z)\)
- Any point on the ZX-plane has \(y=0\), i.e., \((x,0,z)\)
Example 1
Name the octants in which the following points lie: \((1,2,3)\), \((4,-2,3)\), \((4,-2,-5)\), \((4,2,-5)\), \((-4,2,-5)\), \((-4,2,5)\), \((-3,-1,6)\), \((-2,-4,-7)\).
Using the sign chart above:
\((1,2,3)\) → Octant I; \((4,-2,3)\) → IV; \((4,-2,-5)\) → VIII; \((4,2,-5)\) → V;
\((-4,2,-5)\) → VI; \((-4,2,5)\) → II; \((-3,-1,6)\) → III; \((-2,-4,-7)\) → VII.
Example 2
Find the image of the point \((-2,3,4)\) in the YZ-plane.
Reflection in the YZ-plane (\(x=0\)) changes the sign of \(x\) only. Image \(=(2,3,4)\).
Example 3
A point is on the \(x\)-axis. What are its \(y\)- and \(z\)-coordinates?
Both \(y=0\) and \(z=0\). The point has the form \((a,0,0)\).
Activity: Build Your Own 3D Axes
L3 ApplyMaterials: Three thin straws/sticks, modelling clay, a protractor, small paper flags (labels), a marble.
Predict: If you place the marble so that its \(x\), \(y\) and \(z\) distances from the corner (origin) are all equal, where will it lie?
- Fix three straws into a ball of clay so that they meet at right angles — label them X, Y, Z.
- Use the protractor to verify each pair makes a \(90°\) angle.
- Mark unit ticks (1 cm apart) on each straw.
- Place the marble at coordinates \((2,3,4)\) using a small stand — count 2 ticks along X, 3 along Y, rise 4 along Z.
- Now identify the octant for \((2,3,4)\), \((-1,2,3)\), and \((0,0,5)\) by pointing to regions of your model.
If \(x=y=z\) (all positive), the marble lies on the line \(x=y=z\), the diagonal of octant I — the space diagonal of a unit cube. You will also notice that \((0,0,5)\) lies exactly on the Z-axis (not inside any octant).
Competency-Based Questions
Scenario: An air-traffic control tower at airport \(O\) uses three reference axes — east along \(x\), north along \(y\), and vertical along \(z\) (all in kilometres). Aircraft \(A\) is tracked at \((6,8,3)\), helicopter \(H\) at \((-2,5,1)\), and drone \(D\) at \((3,-4,2)\).
Q1. In which octant does the helicopter \(H(-2,5,1)\) lie?
L3 ApplyAnswer: (b) Octant II. Signs are \((-, +, +)\), matching the second octant.
Q2. Analyse: if the drone's altitude drops to ground level, what are its new coordinates, and on which coordinate plane does it now lie?
L4 AnalyseAnswer: Ground level means \(z=0\). New position: \((3,-4,0)\). Since \(z=0\), the drone lies on the XY-plane.
Q3. Evaluate: a radar claims all three aircraft are collinear (lie on a single straight line through \(O\)). Is this claim correct?
L5 EvaluateAnswer: For points to be collinear with \(O\), the ratios \(x:y:z\) must be equal. For \(A\): \(6:8:3\); for \(H\): \(-2:5:1\); for \(D\): \(3:-4:2\). These ratios are all different, so the three aircraft are not collinear. The claim is false.
Q4. Create a 4-point flight path beginning at origin, passing through exactly two different octants (I and V), and ending at altitude \(z=0\). Give explicit coordinates.
L6 CreateSample Answer: \(O(0,0,0) \to P_1(2,3,4)\) [Octant I] \(\to P_2(5,6,-2)\) [Octant V, since \(z<0, x>0, y>0\)] \(\to P_3(4,3,0)\) [on XY-plane]. The path visits Octants I and V and ends at ground level. Many valid answers exist.
Assertion–Reason Questions
Assertion (A): The point \((0,-3,5)\) lies in the YZ-plane.
Reason (R): A point lies in the YZ-plane if and only if its \(x\)-coordinate is zero.
Reason (R): A point lies in the YZ-plane if and only if its \(x\)-coordinate is zero.
Answer: (a). A is true (since \(x=0\)). R correctly states the defining condition, so R explains A.
Assertion (A): The point \((4,-2,-5)\) lies in octant VIII.
Reason (R): Octant VIII is the region where \(x>0\), \(y<0\), \(z<0\).
Reason (R): Octant VIII is the region where \(x>0\), \(y<0\), \(z<0\).
Answer: (a). Signs \((+,-,-)\) match octant VIII; R is the definition, so it explains A.
Assertion (A): Every point of the \(x\)-axis lies in the XY-plane.
Reason (R): A point lies in the XY-plane if and only if its \(z\)-coordinate is zero.
Reason (R): A point lies in the XY-plane if and only if its \(z\)-coordinate is zero.
Answer: (a). Points of the \(x\)-axis have form \((a,0,0)\), hence \(z=0\), hence lie in XY-plane. R explains A.
Frequently Asked Questions
How many octants are there in 3D space?
The three coordinate planes divide space into 8 regions called octants. The one with all positive coordinates is the first octant.
What are the three coordinate planes?
The xy-plane (z = 0), yz-plane (x = 0), and zx-plane (y = 0). Each pair of axes forms one coordinate plane.
How do you locate a point P(x, y, z) in 3D?
From origin O move x units along the x-axis, then y units parallel to the y-axis, then z units parallel to the z-axis. The resulting point is P.
What are the coordinates of the origin?
The origin has coordinates (0, 0, 0). All three axes meet at the origin.
What is a right-handed coordinate system?
Using the right hand: index finger along positive x-axis, middle finger along positive y-axis and thumb along positive z-axis - this is a right-handed 3D system.
How is 3D geometry different from 2D?
2D uses two coordinates and one plane; 3D uses three coordinates, three mutually perpendicular axes, three coordinate planes, and eight octants - accommodating depth as well.
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