H2 Maths Notes (JC 1-2): 3.2) Scalar and Vector Products

Before you revise\ Keep a checklist of formulas: dot product, projection, cross product determinant, and triple scalar product. Write them out at the top of every practice set until they become automatic.

Scalar (Dot) Product

• Definition: \( \vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 = \lVert \vec{a} \rVert \lVert \vec{b} \rVert \cos\theta \).
• Use to compute angles or test perpendicularity.
• Projection of \( \vec{a} \) onto \( \vec{b} \): \( \operatorname{proj}_{\vec{b}} \vec{a} = \frac{\vec{a} \cdot \vec{b}}{\lVert \vec{b} \rVert^2} \vec{b} \).

Example -- Projection length

Given \( \vec{a} = \begin{pmatrix} 3 \ -2 \ 1 \end{pmatrix} \) and \( \vec{b} = \begin{pmatrix} 1 \ 4 \ -2 \end{pmatrix} \), \[ \operatorname{proj}_{\vec{b}} \vec{a} = \frac{3 \times 1 + (-2) \times 4 + 1 \times (-2)}{1^2 + 4^2 + (-2)^2} \begin{pmatrix} 1 \ 4 \ -2 \end{pmatrix} = \frac{-7}{21} \begin{pmatrix} 1 \ 4 \ -2 \end{pmatrix}. \]

Vector (Cross) Product

• \( \vec{a} \times \vec{b} \) is perpendicular to both \( \vec{a} \) and \( \vec{b} \) with magnitude \( \lVert \vec{a} \rVert \lVert \vec{b} \rVert \sin\theta \).
• Determinant form: \[ \vec{a} \times \vec{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}. \]
• Result vector orientation follows the right-hand rule.

Example -- Cross product

Take \( \vec{a} = \begin{pmatrix} 2 \ 1 \ -1 \end{pmatrix} \) and \( \vec{b} = \begin{pmatrix} -3 \ 0 \ 4 \end{pmatrix} \). \[ \vec{a} \times \vec{b} = \begin{pmatrix} 1 \times 4 - (-1) \times 0 \ -\bigl(2 \times 4 - (-1) \times (-3)\bigr) \ 2 \times 0 - 1 \times (-3) \end{pmatrix} = \begin{pmatrix} 4 \ -5 \ 3 \end{pmatrix}. \]

Areas and Volumes

• Area of parallelogram spanned by \( \vec{a}, \vec{b} \): \( \lVert \vec{a} \times \vec{b} \rVert \).
• Area of triangle: half the parallelogram area.
• Volume of parallelepiped: \( \lvert \vec{a} \cdot (\vec{b} \times \vec{c}) \rvert \).

Example -- Volume

Let \( \vec{a} = \begin{pmatrix} 1 \ 2 \ 0 \end{pmatrix} \, \vec{b} = \begin{pmatrix} -1 \ 3 \ 1 \end{pmatrix} \, \vec{c} = \begin{pmatrix} 2 \ 0 \ 4 \end{pmatrix} \). \[ \vec{b} \times \vec{c} = \begin{pmatrix} 12 \ 6 \ -6 \end{pmatrix}, \quad \vec{a} \cdot (\vec{b} \times \vec{c}) = 12 + 12 + 0 = 24. \] Volume \( = \lvert 24 \rvert = 24 \) cubic units.

Coplanarity and Shortest Distance

• Three vectors coplanar if scalar triple product is zero.
• Distance from point to line: project difference vector onto perpendicular direction:
  \[ d = \frac{\lVert (\vec{OP} - \vec{OA}) \times \vec{d} \rVert}{\lVert \vec{d} \rVert}. \]
• Distance between skew lines uses cross product of direction vectors.

Calculator Workflow

• Use GC matrix operations to compute determinants efficiently.
• Some calculators offer direct dotP / crossP; if not, store components and evaluate manually.
• Verify perpendicularity by checking \( \vec{a} \cdot (\vec{a} \times \vec{b}) = 0 \).

Exam Watch Points

• Maintain exact forms (square roots) in intermediate steps; round only at the end.
• When describing vector geometry, supplement algebra with a labelled diagram.
• Clearly state direction vectors, normals, and parameters when concluding lines or planes are perpendicular/parallel.
• Double-check sign conventions in determinant expansion to avoid transcription errors.

Quick Revision Checklist

• [ ] Compute dot products, cross products, and projections with confidence.
• [ ] Apply vector product results to area, volume, and distance problems.
• [ ] Diagnose coplanarity and perpendicularity quickly using scalar triple products.
• [ ] Explain geometric interpretations (angles, shortest distances) in full sentences.

Next steps: Tackle Topic 3.3 for vector equations of planes and 3D intersections.