H2 Maths Notes (JC 1-2): 3.3) Three-Dimensional Vector Geometry

Before you revise\ Sketch every 3D configuration with axes labelled. State direction vectors, normal vectors, and parameters clearly to avoid mixing up lines and planes in algebraic working.

Plane Equations

• Vector form: \( \vec{r} = \vec{OA} + s \vec{u} + t \vec{v} \) with non-parallel \( \vec{u}, \vec{v} \).
• Scalar form: \( \vec{n} \cdot (\vec{r} - \vec{OA}) = 0 \) where \( \vec{n} = \vec{u} \times \vec{v} \).
• Cartesian: \( ax + by + cz = d \) obtained by expanding dot product.

Example -- Plane through three points

Points \( A(1, 0, 2) \, B(3, -1, 4) \, C(2, 2, 0) \).

1. Direction vectors: \( \vec{AB} = \begin{pmatrix} 2 \ -1 \ 2 \end{pmatrix} \, \vec{AC} = \begin{pmatrix} 1 \ 2 \ -2 \end{pmatrix} \).
2. Normal: \( \vec{n} = \vec{AB} \times \vec{AC} = \begin{pmatrix} -2 \ 6 \ 5 \end{pmatrix} \).
3. Plane equation: \( -2(x - 1) + 6(y - 0) + 5(z - 2) = 0 \Rightarrow 2x - 6y - 5z + 8 = 0 \).

Intersections

• Line-plane: substitute line equation into plane; solve for parameter to find intersection point.
• Plane-plane: solve simultaneous equations or express intersection line as \( \vec{r} = \vec{OP} + \lambda \vec{d} \) where \( \vec{d} = \vec{n}_1 \times \vec{n}_2 \).
• Line-line: solve parameter system; if no common solution and lines not parallel, they are skew.

Example -- Line-plane intersection

Line \( \vec{r} = \begin{pmatrix} -1 \ 2 \ 3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \ -1 \ 4 \end{pmatrix} \) and plane \( 3x - 2y + z = 7 \).

1. Substitute \( x = -1 + 2\lambda \, y = 2 - \lambda \, z = 3 + 4\lambda \).
2. Equation: \( 3(-1 + 2\lambda) - 2(2 - \lambda) + (3 + 4\lambda) = 7 \).
3. Simplify: \( -3 + 6\lambda - 4 + 2\lambda + 3 + 4\lambda = 7 \) giving \( 12\lambda - 4 = 7 \Rightarrow \lambda = \tfrac{11}{12} \).
4. Intersection point: \( \vec{r} = \begin{pmatrix} \tfrac{5}{6} \ \tfrac{13}{12} \ \tfrac{20}{3} \end{pmatrix} \).

Angles Between Lines and Planes

• Angle between two lines: use dot product of direction vectors.
• Angle between line and plane: use complement of angle between line direction and plane normal.
• Angle between planes: use dot product of normals.

Example -- Line-plane angle

Line direction \( \vec{d} = \begin{pmatrix} 1 \ 2 \ -2 \end{pmatrix} \), plane normal \( \vec{n} = \begin{pmatrix} 2 \ -1 \ 2 \end{pmatrix} \). \[ \cos\theta = \frac{\lvert \vec{d} \cdot \vec{n} \rvert}{\lVert \vec{d} \rVert \lVert \vec{n} \rVert} = \frac{\lvert 1\times 2 + 2 \times (-1) + (-2) \times 2 \rvert}{\sqrt{1^2 + 2^2 + (-2)^2} \sqrt{2^2 + (-1)^2 + 2^2}} = \frac{6}{3 \sqrt{9}} = \frac{2}{3}. \] Angle between line and plane is \( 90^\circ - \arccos\bigl( \tfrac{2}{3} \bigr) \).

Distances in 3D

• Point to plane distance: \ \[ d = \frac{\lvert \vec{n} \cdot (\vec{OP} - \vec{OA}) \rvert}{\lVert \vec{n} \rVert}. \]
• Skew lines: find vector between points on each line and resolve perpendicular component using cross product magnitude divided by \( \lVert \vec{d}_1 \times \vec{d}_2 \rVert \).

Geometric Proofs

• To prove points coplanar, show scalar triple product zero or express one vector as combination of others.
• For perpendicularity, show dot product zero between appropriate direction/normal vectors.
• For parallel planes, normals are proportional; for coincidence, also verify one point satisfies both equations.

Calculator Workflow

• Use GC matrix solver for simultaneous equations (plane intersections).
• Store normals and direction vectors to reuse in dot/cross product calculations.
• When solving distances, keep expressions exact; use sqrt( only at final stage if decimals required.

Exam Watch Points

• Label all parameters \( \lambda, \mu, s, t \) clearly to avoid confusion.
• State final answers in exact form where possible and include units if the context demands.
• Support algebraic conclusions with geometric language (“The line intersects the plane at…”).
• When planes are perpendicular, state that normals are perpendicular and compute the dot product to confirm.

Quick Revision Checklist

• [ ] Convert between vector/Cartesian forms of planes confidently.
• [ ] Solve line-plane and plane-plane intersections systematically.
• [ ] Compute angles and distances between lines/planes with correct formulae.
• [ ] Justify geometric relationships (parallel, perpendicular, coplanar) with vector reasoning.

Next steps: Begin Topic 4.1 for complex numbers and Argand diagrams.