Introduction to Three Dimensional Geometry Class 11: notes and important questions

Introduction to Three Dimensional Geometry: revision notes

1. Axes, planes and octants

Three mutually perpendicular lines through the origin \(O\) are the \(x\)-, \(y\)- and \(z\)-axes. They determine three coordinate planes: the \(xy\)-plane (\(z = 0\)), the \(yz\)-plane (\(x = 0\)) and the \(zx\)-plane (\(y = 0\)). The planes divide space into eight octants, numbered I to VIII: I \((+, +, +)\), II \((-, +, +)\), III \((-, -, +)\), IV \((+, -, +)\), and V to VIII the same with \(z \lt 0\).

2. Coordinates of a point

\(P(x, y, z)\): \(x\), \(y\), \(z\) are the signed distances of \(P\) from the \(yz\)-, \(zx\)- and \(xy\)-planes.

• On the \(x\)-axis: \((x, 0, 0)\); on the \(y\)-axis: \((0, y, 0)\); on the \(z\)-axis: \((0, 0, z)\).
• In the \(xy\)-plane: \((x, y, 0)\), and so on. The foot of the perpendicular from \((a, b, c)\) to the \(xy\)-plane is \((a, b, 0)\).

3. Distance between two points

\[PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]

• From the origin: \(\sqrt{x^2 + y^2 + z^2}\).
• Distance of \((a, b, c)\) from the \(x\)-axis: \(\sqrt{b^2 + c^2}\) (from the foot \((a, 0, 0)\)); from the \(xy\)-plane: \(|c|\).

4. Using the distance formula

• Collinear points: \(AB + BC = AC\) for the right order.
• Isosceles / equilateral / right triangles, parallelograms and rectangles: compare sides (and diagonals).
• Equidistant points and loci: set squared distances equal; the points at distance \(r\) from \((a, b, c)\) satisfy \((x - a)^2 + (y - b)^2 + (z - c)^2 = r^2\).

Worked example 1

Find the distance between \((2, -1, 3)\) and \((-2, 1, 3)\).

\(\sqrt{16 + 4 + 0} = \sqrt{20} = 2\sqrt5\).

Worked example 2

Name the octant of \((4, -2, -3)\).

Signs \((+, -, -)\): octant VIII.

Worked example 3

Find the point on the \(y\)-axis equidistant from \((1, 1, 0)\) and \((0, 3, 1)\).

\((0, y, 0)\): \(1 + (y - 1)^2 = (y - 3)^2 + 1 \Rightarrow 4y = 8\), so \((0, 2, 0)\).

Common errors

• Forgetting the \(z\) term in the distance formula.
• Distance from the \(x\)-axis taken as \(|a|\): it is \(\sqrt{b^2 + c^2}\).
• Confusing "on the \(x\)-axis" \((x, 0, 0)\) with "in the \(xy\)-plane" \((x, y, 0)\).
• Claiming collinearity from two equal distances instead of \(AB + BC = AC\).

Exam tips

• Write the differences in brackets first, \((4 - 1)^2 + (2 - (-2))^2 + \ldots\), then square.
• For "show that", end with the relation you proved (for example \(AB^2 + AC^2 = BC^2\)).